Example 5-1 Mobile Response Time The response time is the speed of page downloads and it is critical for a mobile Web site. As the response time increases, customers become more frustrated and potentially abandon the site for a competitive one. Let X denote the number of bars of service, and let Y denote the response time (to the nearest second) for a particular user and site.

FIGURE 5-1 Joint probability distribution of X and Y in Example 5-1.

By specifying the probability of each of the points in Fig. 5-1, we specify the joint probability distribution of X and Y. Similarly to an individual random variable, we define the range of the random variables (X,Y) to be the set of points (x, y) in two-dimensional space for which the probability that X = x and Y = y is positive.

Example 5-2 Server Access Time Let the random variable X denote the time until a computer server connects to your machine (in milliseconds), and let Y denote the time until the server authorizes you as a valid user (in milliseconds). Each of these random variables measures the wait from a common starting time and X<Y. Assume that the joint probability density function for X and Y is

Reasonable assumptions can be used to develop such a distribution, but for now, our focus is on only the joint probability density function.

The region with nonzero probability is shaded in Fig. 5-4. The property that this joint probability density function integrates to 1 can be verified by the integral of f_XY(x, y) over this region as follows:

FIGURE 5-4 The joint probability density function of X and Y is nonzero over the shaded region.

FIGURE 5-5 Region of integration for the probability that X<1000 and Y <2000 is darkly shaded.

The probability that X<1000 and Y<2000 is determined as the integral over the darkly shaded region in Fig. 5-5.

Practical Interpretation: A joint probability density function enables probabilities for two (or more) random variables to be calculated as in these examples.

Example 5-3 Marginal Distribution The joint probability distribution of X and Y in Fig. 5-1 can be used to find the marginal probability distribution of X. For example,

The marginal probability distribution for X is found by summing the probabilities in each column whereas the marginal probability distribution for Y is found by summing the probabilities in each row. The results are shown in Fig. 5-6.

Example 5-4 Server Access Time For the random variables that denote times in Example 5-2, calculate the probability that Y exceeds 2000 milliseconds.

This probability is determined as the integral of f_XY(x,y) over the darkly shaded region in Fig. 5-7. The region is partitioned into two parts and different limits of integration are determined for each part.

The first integral is

The second integral is

FIGURE 5-7 Region of integration for the probability that Y > 2000 is darkly shaded, and it is partitioned into two regions with x <2000 and x >2000.

Therefore,

Alternatively, the probability can be calculated from the marginal probability distribution of Y as follows. For y > 0,

We have obtained the marginal probability density function of Y. Now,

Example 5-5 Conditional Probabilities for Mobile Response Time For Example 5-1, X and Y denote the number of bars of signal strength and response time, respectively. Then,

The probability that Y = 2 given that X = 3 is

Further work shows that

and

Note that P(Y = 1|X = 3) + P(Y = 2|X = 3) + P(Y = 3|X = 3) + P(Y = 4|X = 3) = 1. This set of probabilities defines the conditional probability distribution of Y given that X = 3.

Example 5-6 Conditional Probability For the random variables that denote times in Example 5-2, determine the conditional probability density function for Y given that X = x.

First the marginal density function of x is determined. For x>0,

This is an exponential distribution with λ = 0.003. Now for 0 < x and x<y, the conditional probability density function is

The conditional probability density function of Y, given that X = 1500, is nonzero on the solid line in Fig. 5-8.

Determine the probability that Y exceeds 2000, given that x = 1500. That is, determine P(Y > 2000|X = 1500).

The conditional probability density function is integrated as follows:

Example 5-7 For the joint probability distribution in Fig. 5-1, f_Y|x(y) is found by dividing each f_XY(x,y) by f_x(x). Here, f_x(x) is simply the sum of the probabilities in each column of Fig. 5-1. The function f_Y|x(y) is shown in Fig. 5-9. In Fig. 5-9, each column sums to 1 because it is a probability distribution.

Properties of random variables can be extended to a conditional probability distribution of Y given X = x. The usual formulas for mean and variance can be applied to a conditional probability density or mass function.

Example 5-8 Conditional Mean And Variance For the random variables that denote times in Example 5-2, determine the conditional mean for Y given that x = 1500.

The conditional probability density function for Y was determined in Example 5-6. Because f_Y|1500(y) is nonzero for y > 1500,

Integrate by parts as follows:

With the constant 0.002e³ reapplied,

Practical Interpretation: If the connect time is 1500 ms, then the expected time to be authorized is 2000 ms.

Example 5-9 For the discrete random variables in Example 5-1, the conditional mean of Y given X = 1 is obtained from the conditional distribution in Fig. 5-9:

The conditional mean is interpreted as the expected response time given that one bar of signal is present. The conditional variance of Y given X = 1 is

Example 5-10 Independent Random Variables An orthopedic physician's practice considers the number of errors in a bill and the number of X-rays listed on the bill. There may or may not be a relationship between these random variables. Let the random variables X and Y denote the number of errors and the number of X-rays on a bill, respectively.

Assume that the joint probability distribution of X and Y is defined by f_XY(x,y) in Fig. 5-10(a). The marginal probability distributions of X and Y are also shown in Fig. 5-10(a). Note that

The conditional probability mass function f_Y|x(y) is shown in Fig. 5-10(b). Notice that for any x, f_Y|x(y) = f_Y(y). That is, knowledge of whether or not the part meets color specifications does not change the probability that it meets length specifications.

Example 5-11 Independent Random Variables Suppose that Example 5-2 is modified so that the joint probability density function of X and Y is f_XY(x,y) = 2×10⁻⁶exp(−0.001x −0.002y) for x ≥ 0 and y ≥ 0. Show that X and Y are independent and determine P(X > 1000, Y < 1000).

Note that the range of positive probability is rectangular so that independence is possible but not yet demonstrated. The marginal probability density function of X is

The marginal probability density function of Y is

Therefore, f_XY(x,y) = f_X(x)f_Y(y) for all x and y, and X and Y are independent.

To determine the probability requested, property (4) of Equation 5-7 can be applied along with the fact that each random variable has an exponential distribution. Therefore,

Example 5-12 Machined Dimensions Let the random variables X and Y denote the lengths of two dimensions of a machined part, respectively. Assume that X and Y are independent random variables, and further assume that the distribution of X is normal with mean 10.5 millimeters and variance 0.0025 (mm²) and that the distribution of Y is normal with mean 3.2 millimeters and variance 0.0036 (mm²). Determine the probability that 10.4 < X<10.6 and 3.15 < Y<3.25.

Because X and Y are independent,

where Z denotes a standard normal random variable.

Practical Interpretation: If random variables are independent, probabilities for multiple variables are often much easier to compute.

Example 5-13 Machined Dimensions Many dimensions of a machined part are routinely measured during production. Let the random variables, X₁, X₂, X₃, and X₄ denote the lengths of four dimensions of a part. Then at least four random variables are of interest in this study.

Example 5-14 Component Lifetimes In an electronic assembly, let the random variables X₁, X₂, X₃, X₄ denote the lifetime of four components, respectively, in hours. Suppose that the joint probability density function of these variables is

What is the probability that the device operates for more than 1000 hours without any failures? The requested probability is P(X₁ > 1000, X₂ > 1000, X₃ > 1000, X₄ > 1000), which equals the multiple integral of f_X1X2X3X4(x₁, x₂, x₃, x₄) over the region x₁ > 1000, x₂ > 1000, x₃ > 1000, x₄ > 1000. The joint probability density function can be written as a product of exponential functions, and each integral is the simple integral of an exponential function. Therefore,

Example 5-15 Probability as a Ratio of Volumes Suppose that the joint probability density function of the continuous random variables X and Y is constant over the region x² + y² ≤ 4. Determine the probability that X² + Y² ≤ 1.

The region that receives positive probability is a circle of radius 2. Therefore, the area of this region is 4π. The area of the region x² + y² ≤ 1 is π. Consequently, the requested probability is π/(4π) = 1/4.

Example 5-16 Points that have positive probability in the joint probability distribution of three random variables X₁, X₂, X₃ are shown in Fig. 5-11. Suppose the 10 points are equally likely with probability 0.1 each. The range is the non-negative integers with x₁ + x₂ + x₃ = 3. The marginal probability distribution of X₂ is found as follows.

Also, E(X₂) = 0(0.4)+1(0.3)+2(0.2)+3(0.1) = 1

Example 5-17 In Chapter 3, we showed that a negative binomial random variable with parameters p and r can be represented as a sum of r geometric random variables X₁, X₂,..., X_r. Each geometric random variable represents the additional trials required to obtain the next success. Because the trials in a binomial experiment are independent, X₁, X₂,..., X_r are independent random variables.

Example 5-18 Layer Thickness Suppose that X₁, X₂, and X₃ represent the thickness in micrometers of a substrate, an active layer, and a coating layer of a chemical product, respectively. Assume that X₁, X₂, and X₃ are independent and normally distributed with μ₁ = 10000, μ₂ = 1000, μ₃ = 80, σ₁ = 250, σ₂ = 20, and σ₃ = 4, respectively. The specifications for the thickness of the substrate, active layer, and coating layer are 9200 < x₁ < 10,800,950 < x₂ < 1050, and 75 < x₃ < 85, respectively. What proportion of chemical products meets all thickness specifications? Which one of the three thicknesses has the least probability of meeting specifications?

The requested probability is P(9200<X₁<10,800, 950<X₂<1050, 75<X₃<85). Because the random variables are independent,

After standardizing, the above equals

where Z is a standard normal random variable. From the table of the standard normal distribution, the requested probability equals

The thickness of the coating layer has the least probability of meeting specifications. Consequently, a priority should be to reduce variability in this part of the process.

   EXERCISES FOR SECTION 5-1

Problem available in WileyPLUS at instructor's discretion.

Go Tutorial Tutoring problem available in WileyPLUS at instructor's discretion.

5-1. Show that the following function satisfies the properties of a joint probability mass function.

Determine the following:

(a) P(X < 2.5, Y < 3)

(b) P(X < 2.5)

(c) P(Y < 3)

(d) P(X > 1.8, Y > 4.7)

(e) E(X), E(Y), V(X), and V(Y).

(f) Marginal probability distribution of X

(g) Conditional probability distribution of Y given that X = 1.5

(h) Conditional probability distribution of X given that Y = 2

(i) E(Y|X = 1.5)

(j) Are X and Y independent?

5-2. Determine the value of c that makes the function f(x,y) = c(x+y) a joint probability mass function over the nine points with x = 1,2,3 and y = 1,2,3.

Determine the following:

(a) P(X = 1, Y < 4)

(b) P(X = 1)

(c) P(Y = 2)

(d) P(X < 2, Y < 2)

(e) E(X), E(Y), V(X), and V(Y)

(f) Marginal probability distribution of X

(g) Conditional probability distribution of Y given that X = 1

(h) Conditional probability distribution of X given that Y = 2

(i) E(Y|X = 1)

(j) Are X and Y independent?

5-3. Show that the following function satisfies the properties of a joint probability mass function.

Determine the following:

(a) P(X < 0.5, Y < 1.5)

(b) P(X < 0.5)

(c) P(Y < 1.5)

(d) P(X > 0.25, Y < 4.5)

(e) E(X), E(Y), V(X), and V(Y)

(f) Marginal probability distribution of X

(g) Conditional probability distribution of Y given that X = 1

(h) Conditional probability distribution of X given that Y = 1

(i) E(X|y = 1)

(j) Are X and Y independent?

5-4. Four electronic printers are selected from a large lot of damaged printers. Each printer is inspected and classified as containing either a major or a minor defect. Let the random variables X and Y denote the number of printers with major and minor defects, respectively. Determine the range of the joint probability distribution of X and Y.

5-5. In the transmission of digital information, the probability that a bit has high, moderate, and low distortion is 0.01, 0.04, and 0.95, respectively. Suppose that three bits are transmitted and that the amount of distortion of each bit is assumed to be independent. Let X and Y denote the number of bits with high and moderate distortion out of the three, respectively. Determine:

(a) f_XY(x, y)

(b) f_X(x)

(c) E(X)

(d) f_Y|1(y)

(e) E(Y|X = 1)

(f) Are X and Y independent?

5-6. A small-business Web site contains 100 pages and 60%, 30%, and 10% of the pages contain low, moderate, and high graphic content, respectively. A sample of four pages is selected without replacement, and X and Y denote the number of pages with moderate and high graphics output in the sample. Determine:

(a) f_XY(x, y)

(b) f_X(x)

(c) E(X)

(d) f_Y|3(y)

(e) E(Y|X = 3)

(f) V(Y|X = 3)

(g) Are X and Y independent?

5-7. A manufacturing company employs two devices to inspect output for quality control purposes. The first device is able to accurately detect 99.3% of the defective items it receives, whereas the second is able to do so in 99.7% of the cases. Assume that four defective items are produced and sent out for inspection. Let X and Y denote the number of items that will be identified as defective by inspecting devices 1 and 2, respectively. Assume that the devices are independent. Determine:

(a) f_XY(x, y)

(b) f_X(x)

(c) E(X)

(d) f_Y|2 (y)

(e) E(Y|X = 2)

(f) V(Y|X = 2)

(g) Are X and Y independent?

5-8. Suppose that the random variables X, Y, and Z have the following joint probability distribution.

Determine the following:

(a) P(X = 2)

(b) P(X = 1, Y = 2)

(c) P(Z < 1.5)

(d) P(X = 1 or Z = 2)

(e) E(X)

(f) P(X = 1|Y = 1)

(g) P(X = 1, Y = 1|Z = 2)

(h) P(X = 1|Y = 1, Z = 2)

(i) Conditional probability distribution of X given that Y = 1 and Z = 2

5-9. An engineering statistics class has 40 students; 60% are electrical engineering majors, 10% are industrial engineering majors, and 30% are mechanical engineering majors. A sample of four students is selected randomly without replacement for a project team. Let X and Y denote the number of industrial engineering and mechanical engineering majors, respectively. Determine the following:

(a) f_XY(x, y)

(b) f_X(x)

(c) E(X)

(d) f_Y|3(y)

(e) E(Y|X = 3)

(f) V(Y|X = 3)

(g) Are X and Y independent?

5-10. An article in the Journal of Database Management [“Experimental Study of a Self-Tuning Algorithm for DBMS Buffer Pools” (2005, Vol. 16, pp. 1–20)] provided the workload used in the TPC-C OLTP (Transaction Processing Performance Council's Version C On-Line Transaction Processing) benchmark, which simulates a typical order entry application. See the following table. The frequency of each type of transaction (in the second column) can be used as the percentage of each type of transaction. Let X and Y denote the average number of selects and updates operations, respectively, required for each type transaction. Determine the following:

(a) P(X < 5)

(b) E(X)

(c) Conditional probability mass function of X given Y = 0

(d) P(X < 6|Y = 0)

(e) E(X|Y = 0)

5-11. For the Transaction Processing Performance Council's benchmark in Exercise 5-10, let X, Y, and Z denote the average number of selects, updates, and inserts operations required for each type of transaction, respectively. Calculate the following:

(a) f_XYZ(x, y, z)

(b) Conditional probability mass function for X and Y given Z = 0

(c) P(X < 6, Y < 6|Z = 0)

(d) E(X|Y = 0, Z = 0)

5-12. In the transmission of digital information, the probability that a bit has high, moderate, or low distortion is 0.01, 0.04, and 0.95, respectively. Suppose that three bits are transmitted and that the amount of distortion of each bit is assumed to be independent. Let X and Y denote the number of bits with high and moderate distortion of the three transmitted, respectively. Determine the following:

Average Frequencies and Operations in TPC-C

(a) Probability that two bits have high distortion and one has moderate distortion

(b) Probability that all three bits have low distortion

(c) Probability distribution, mean, and variance of X

(d) Conditional probability distribution, conditional mean, and conditional variance of X given that Y = 2

5-13. Determine the value of c such that the function f(x,y) = cxy for 0 < x < 3 and 0 < y < 3 satisfies the properties of a joint probability density function.

Determine the following:

(a) P(X < 2, Y < 3)

(b) P(X < 2.5)

(c) P(1 < Y < 2.5)

(d) P(X > 1.8, 1 < Y < 2.5)

(e) E(X)

(f) P(X < 0, Y < 4)

(g) Marginal probability distribution of X

(h) Conditional probability distribution of Y given that X = 1.5

(i) E(Y|X) = 1.5)

(j) P(Y < 2|X = 1.5)

(k) Conditional probability distribution of X given that Y = 2

5-14. Determine the value of c that makes the function f(x, y) = c(x + y) a joint probability density function over the range 0 < x < 3 and x < y < x + 2.

Determine the following:

(a) P(X < 1, Y < 2)

(b) P(1 < X < 2)

(c) P(Y > 1)

(d) P(X < 2, Y < 2)

(e) E(X)

(f) V(X)

(g) Marginal probability distribution of X

(h) Conditional probability distribution of Y given that X = 1

(i) E(Y|X = 1)

(j) P(Y > 2|X = 1)

(k) Conditional probability distribution of X given that Y = 2

5-15. Determine the value of c that makes the function f(x,y) = c(x + y) a joint probability density function over the range 0 < x<3 and 0 < y<x.

Determine the following:

(a) P(X < 1,Y < 2)

(b) P(1 < X < 2)

(c) P(Y > 1)

(d) P(X < 2, Y < 2)

(e) E(X)

(f) E(Y)

(g) Marginal probability distribution of X

(h) Conditional probability distribution of Y given X = 1

(i) E(Y|X = 1)

(j) P(Y > 2|X = 1)

(k) Conditional probability distribution of X given Y = 2

5-16. Determine the value of c that makes the function f(x, y) = ce^{− 2x −3y} a joint probability density function over the range 0 < x and 0 < y<x.

Determine the following:

(a) P(X < 1, Y < 2)

(b) P(1 < X < 2)

(c) P(Y > 3)

(d) P(X < 2, Y < 2)

(e) E(X)

(f) E(Y)

(g) Marginal probability distribution of X

(h) Conditional probability distribution of Y given X = 1

(i) E(Y|X = 1)

(j) Conditional probability distribution of X given Y = 2

5-17. Determine the value of c that makes the function f(x,y) = ce^−2x−3y, a joint probability density function over the range 0 < x and x<y.

Determine the following:

(a) P(X < 1, Y < 2)

(b) P(1 < X < 2)

(c) P(Y > 3)

(d) P(X < 2, Y < 2)

(e) E(X)

(f) E(Y)

(g) Marginal probability distribution of X

(h) Conditional probability distribution of Y given X = 1

(i) E(Y|X = 1)

(j) P(Y < 2|X = 1)

(k) Conditional probability distribution of X given Y = 2

5-18. The conditional probability distribution of Y given X = x is f_Y|x(y) = xe^−xy for y > 0, and the marginal probability distribution of X is a continuous uniform distribution over 0 to 10.

(a) Graph f_Y|x(y) = xe^−xy for y > 0 for several values of x.

Determine:

(b) P(Y < 2|X = 2)

(c) E(Y|X = 2)

(d) E(Y|X = x)

(e) f_XY(x, y)

(f) f_Y(y)

5-19. Two methods of measuring surface smoothness are used to evaluate a paper product. The measurements are recorded as deviations from the nominal surface smoothness in coded units. The joint probability distribution of the two measurements is a uniform distribution over the region 0 < x < 4, 0 < y, and x − 1 < y < x + 1. That is, f_XY(x, y) = c for x and y in the region. Determine the value for c such that f_XY(x, y) is a joint probability density function.

Determine the following:

(a) P(X < 0.5, Y < 0.5)

(b) P(X < 0.5)

(c) E(X)

(d) E(Y)

(e) Marginal probability distribution of X

(f) Conditional probability distribution of Y given X = 1

(g) E(Y|X = 1)

(h) P(Y < 0.5|X = 1)

5-20. The time between surface finish problems in a galvanizing process is exponentially distributed with a mean of 40 hours. A single plant operates three galvanizing lines that are assumed to operate independently.

(a) What is the probability that none of the lines experiences a surface finish problem in 40 hours of operation?

(b) What is the probability that all three lines experience a surface finish problem between 20 and 40 hours of operation?

(c) Why is the joint probability density function not needed to answer the previous questions?

5-21. A popular clothing manufacturer receives Internet orders via two different routing systems. The time between orders for each routing system in a typical day is known to be exponentially distributed with a mean of 3.2 minutes. Both systems operate independently.

(a) What is the probability that no orders will be received in a 5-minute period? In a 10-minute period?

(b) What is the probability that both systems receive two orders between 10 and 15 minutes after the site is officially open for business?

(c) Why is the joint probability distribution not needed to answer the previous questions?

5-22. The blade and the bearings are important parts of a lathe. The lathe can operate only when both of them work properly. The lifetime of the blade is exponentially distributed with the mean three years; the lifetime of the bearings is also exponentially distributed with the mean four years. Assume that each lifetime is independent.

(a) What is the probability that the lathe will operate for at least five years?

(b) The lifetime of the lathe exceeds what time with 95% probability?

5-23. Suppose that the random variables X, Y, and Z have the joint probability density function f(x, y, z) = 8xyz for 0 < x<1, 0 < y<1, and 0 < z<1. Determine the following:

(a) P(X < 0.5)

(b) P(X < 0.5, Y < 0.5)

(c) P(Z < 2)

(d) P(X < 0.5 or Z < 2)

(e) E(X)

(f) P(X < 0.5|Y = 0.5)

(g) P(X < 0.5, Y < 0.5|Z = 0.8)

(h) Conditional probability distribution of X given that Y = 0.5 and Z = 0.8

(i) P(X < 0.5|Y = 0.5, Z = 0.8)

5-24. Suppose that the random variables X, Y, and Z have the joint probability density function f_XYZ(x, y, z) = c over the cylinder x² + y² < 4 and 0 < z < 4. Determine the constant c so that f_XYZ(x, y, z) is a probability density function.

Determine the following:

(a) P(X² + Y² < 2)

(b) P(Z < 2)

(c) E(X)

(d) P(X < 1|Y = 1)

(e) P(X² + Y² < 1|Z = 1)

(f) Conditional probability distribution of Z given that X = 1 and Y = 1.

5-25. Determine the value of c that makes f_XYZ(x, y, z) = c a joint probability density function over the region x > 0, y > 0, z > 0, and x + y + z < 1.

Determine the following:

(a) P(X < 0.5, Y < 0.5, Z < 0.5)

(b) P(X < 0.5, Y < 0.5)

(c) P(X < 0.5)

(d) E(X)

(e) Marginal distribution of X

(f) Joint distribution of X and Y

(g) Conditional probability distribution of X given that Y = 0.5 and Z = 0.5

(h) Conditional probability distribution of X given that Y = 0.5

5-26. The yield in pounds from a day's production is normally distributed with a mean of 1500 pounds and standard deviation of 100 pounds. Assume that the yields on different days are independent random variables.

(a) What is the probability that the production yield exceeds 1400 pounds on each of five days next week?

(b) What is the probability that the production yield exceeds 1400 pounds on at least four of the five days next week?

5-27. The weights of adobe bricks used for construction are normally distributed with a mean of 3 pounds and a standard deviation of 0.25 pound. Assume that the weights of the bricks are independent and that a random sample of 20 bricks is selected.

(a) What is the probability that all the bricks in the sample exceed 2.75 pounds?

(b) What is the probability that the heaviest brick in the sample exceeds 3.75 pounds?

5-28. A manufacturer of electroluminescent lamps knows that the amount of luminescent ink deposited on one of its products is normally distributed with a mean of 1.2 grams and a standard deviation of 0.03 gram. Any lamp with less than 1.14 grams of luminescent ink fails to meet customers' specifications. A random sample of 25 lamps is collected and the mass of luminescent ink on each is measured.

(a) What is the probability that at least one lamp fails to meet specifications?

(b) What is the probability that five or fewer lamps fail to meet specifications?

(c) What is the probability that all lamps conform to specifications?

(d) Why is the joint probability distribution of the 25 lamps not needed to answer the previous questions?

5-29. The lengths of the minor and major axes are used to summarize dust particles that are approximately elliptical in shape. Let X and Y denote the lengths of the minor and major axes (in micrometers), respectively. Suppose that f_X(x) = exp(−x), 0 < x and the conditional distribution f_Y|x(y) = exp[−(y−x)], x < y. Answer or determine the following:

(a) That f_Y|x(y) is a probability density function for any value of x.

(b) P(X < Y) and comment on the magnitudes of X and Y.

(c) Joint probability density function f_XY(x, y).

(d) Conditional probability density function of X given Y = y.

(e) P(Y < 2|X = 1)

(f) E(Y|X = 1)

(g) P(X < 1, Y < 1)

(h) P(Y < 2)

(i) c such that P(Y < c) = 0.9

(j) Are X and Y independent?

5-30. An article in Health Economics [“Estimation of the Transition Matrix of a Discrete-Time Markov Chain” (2002, Vol.11, pp. 33–42)] considered the changes in CD4 white blood cell counts from one month to the next. The CD4 count is an important clinical measure to determine the severity of HIV infections. The CD4 count was grouped into three distinct categories: 0–49, 50–74, and ≥ 75. Let X and Y denote the (category minimum) CD4 count at a month and the following month, respectively. The conditional probabilities for Y given values for X were provided by a transition probability matrix shown in the following table.

This table is interpreted as follows. For example, P(Y = 50|X = 75) = 0.0717. Suppose also that the probability distribution for X is P(X = 75) = 0.9,P(X = 50) = 0.08,P(X = 0) = 0.02. Determine the following:

(a) P(Y ≤ 50|X = 50)

(b) P(X = 0, Y = 75)

(c) E(Y|X = 50)

(d) f_Y(y)

(e) f_XY(x, y)

(f) Are X and Y independent?

5-31. An article in Clinical Infectious Diseases [“Strengthening the Supply of Routinely Administered Vaccines in the United States: Problems and Proposed Solutions” (2006, Vol.42(3), pp. S97–S103)] reported that recommended vaccines for infants and children were periodically unavailable or in short supply in the United States. Although the number of doses demanded each month is a discrete random variable, the large demands can be approximated with a continuous probability distribution. Suppose that the monthly demands for two of those vaccines, namely measles–mumps–rubella (MMR) and varicella (for chickenpox), are independently, normally distributed with means of 1.1 and 0.55 million doses and standard deviations of 0.3 and 0.1 million doses, respectively. Also suppose that the inventory levels at the beginning of a given month for MMR and varicella vaccines are 1.2 and 0.6 million doses, respectively.

(a) What is the probability that there is no shortage of either vaccine in a month without any vaccine production?

(b) To what should inventory levels be set so that the probability is 90% that there is no shortage of either vaccine in a month without production? Can there be more than one answer? Explain.

5-32. The systolic and diastolic blood pressure values (mm Hg) are the pressures when the heart muscle contracts and relaxes (denoted as Y and X, respectively). Over a collection of individuals, the distribution of diastolic pressure is normal with mean 73 and standard deviation 8. The systolic pressure is conditionally normally distributed with mean 1.6x when X = x and standard deviation of 10. Determine the following:

(a) Conditional probability density function f_Y|73(y) of Y given X = 73

(b) P(Y < 115|X = 73)

(c) E(Y|X = 73)

(d) Recognize the distribution f_XY(x, y) and identify the mean and variance of Y and the correlation between X and Y

Example 5-19 Expected Value of a Function of Two Random Variables For the joint probability distribution of the two random variables in Example 5-1, calculate E[(X − μ_X)(Y − μ_Y)].

The result is obtained by multiplying x − μ_X times y − μ_Y, times f_xy(X, Y) for each point in the range of (X, Y). First, μ_X and μ_Y were determined previously from the marginal distributions for X and Y:

and

Therefore,

Example 5-20 In Example 5-1, the random variables X and Y are the number of signal bars and the response time (to the nearest second), respectively. Interpret the covariance between X and Y as positive or negative.

As the signal bars increase, the response time tends to decrease. Therefore, X and Y have a negative covariance. The covariance was calculated to be −0.5815 in Example 5-19.

Example 5-21 Covariance For the discrete random variables X and Y with the joint distribution shown in Fig. 5-13, determine σ_XY and ρ_XY.

The calculations for E(XY), E(X), and V(X) are as follows.

Because the marginal probability distribution of Y is the same as for X, E(Y) = 1.8 and V(Y) = 1.36. Consequently,

Furthermore,

Example 5-22 Correlation Suppose that the random variable X has the following distribution: P(X = 1) = 0.2, P(X = 2) = 0.6, P(X = 3) = 0.2. Let Y = 2X + 5. That is, P(Y = 7) = 0.2, P(Y = 11) = 0.2. Determine the correlation between X and Y. Refer to Fig. 5-14.

Because X and Y are linearly related, ρ = 1. This can be verified by direct calculations: Try it.

Example 5-23 Independence Implies Zero Covariance For the two random variables in Fig. 5-15, show that σ_XY = 0.

The two random variables in this example are continuous random variables. In this case, E(XY) is defined as the double integral over the range of (X,Y). That is,

FIGURE 5-15 Random variables with zero covariance from Example 5-22.

Also,

Thus,

It can be shown that these two random variables are independent. You can check that f_XY(x, y) = f_X(x)f_Y(y) for all x and y.

   EXERCISES FOR SECTION 5-2

Problem available in WileyPLUS at instructor's discretion.

Go Tutorial Tutoring problem available in WileyPLUS at instructor's discretion.

5-33. Determine the covariance and correlation for the following joint probability distribution:

5-34. Determine the covariance and correlation for the following joint probability distribution:

5-35. Determine the value for c and the covariance and correlation for the joint probability mass function f_XY(x, y) = c(x+ y) for x = 1,2,3 and y = 1,2,3.

5-36. Determine the covariance and correlation for the joint proba.bility distribution shown in Fig. 5-10(a) and described in Example 5-10.

5-37. Patients are given a drug treatment and then evaluated. Symptoms either improve, degrade, or remain the same with probabilities 0.4, 0.1, 0.5, respectively. Assume that four independent patients are treated and let X and Y denote the number of patients who improve or degrade. Are X and Y independent? Calculate the covariance and correlation between X and Y.

5-38. For the Transaction Processing Performance Council's benchmark in Exercise 5-10, let X, Y, and Z denote the average number of selects, updates, and inserts operations required for each type of transaction, respectively. Calculate the following:

(a) Covariance between X and Y

(b) Correlation between X and Y

(c) Covariance between X and Z

(d) Correlation between X and Z

5-39. Determine the value for c and the covariance and correlation for the joint probability density function f_XY(x, y) = cxy over the range 0 < x < 3 and 0 < y < x.

5-40. Determine the value for c and the covariance and correlation for the joint probability density function f_XY(x, y) = c over the range 0 < x < 5, 0 < y, and x −1 < y < x +1.

5-41. Determine the covariance and correlation for the joint probability density function f_XY(x, y) = e^−x−y over the range 0 < x and 0 < y.

5-42. Determine the covariance and correlation for the joint probability density function f_XY(x, y) = over the range 0 < x and x<y from Example 5-2.

5-43. The joint probability distribution is

Show that the correlation between X and Y is zero but X and Y are not independent.

5-44. Determine the covariance and correlation for the CD4 counts in a month and the following month in Exercise 5-30.

5-45. Determine the covariance and correlation for the lengths of the minor and major axes in Exercise 5-29.

5-46. Suppose that X and Y are independent continuous random variables. Show that σ_XY = 0.

5-47. Suppose that the correlation between X and Y is ρ. For constants a,b,c, and d, what is the correlation between the random variables U = aX + b and V = cY + d?

Example 5-21 Digital Channel We might be interested in a probability such as the following. Of the 20 bits received, what is the probability that 14 are excellent, 3 are good, 2 are fair, and 1 is poor? Assume that the classifications of individual bits are independent events and that the probabilities of E, G, F, and P are 0.6, 0.3, 0.08, and 0.02, respectively. One sequence of 20 bits that produces the specified numbers of bits in each class can be represented as

Using independence, we find that the probability of this sequence is

Clearly, all sequences that consist of the same numbers of E's, G's, F's, and P's have the same probability. Consequently, the requested probability can be found by multiplying 2.708×10⁻⁹ by the number of sequences with 14 E's, 3 G's, 2 F's, and 1P. The number of sequences is found from Chapter 2 to be

Therefore, the requested probability is

Example 5-36 Moment-Generating Function for a Binomial Random Variable Suppose that X has a binomial distribution, that is

Determine the moment-generating function and use it to verify that the mean and variance of the binomial random variable are μ = np and σ² = np(1 − p).

From the definition of a moment-generatingfunction, we have

This last summation is the binomial expansion of [pe^t + (1 − p)]ⁿ, so

Taking the first and second derivatives, we obtain

and

If we set t = 0 in M′_X(t), we obtain

which is the mean of the binomial random variable X. Now if we set t = 0 in (t),

Therefore, the variance of the binomial random variable is

Example 5-37 Moment-Generating Function for a Normal Random Variable Find the moment-generating function of the normal random variable and use it to show that the mean and variance of this random variable are μ and σ², respectively.

The moment-generating function is

If we complete the square in the exponent, we have

and then

Let u = [x − (μ + tσ²)]/σ. Then dx = σdu, and the last expression above becomes

Now the integral is just the total area under a standard normal density, which is 1, so the moment-generating function of a normal random variable is

Differentiating this function twice with respect to t and setting t = 0 in the result, yields

Therefore, the variance of the normal random variable is

Example 5-38 Distribution of a Sum of Poisson Random Variables Suppose that X₁ and X₂ are two independent Poisson random variables with parameters λ₁ and λ₂, respectively. Determine the probability distribution of Y = X₁ + X₂.

The moment-generating function of a Poisson random variable with parameter λ is

so the moment-generating functions of X₁ and X₂ are M_X1(t) = e^{λ₁(et −1)} and M_X2(t) = e^{λ₂(et − 1)}, respectively. Using Equation 5-35, the moment-generating function of Y = X₁ + X₂ is

which is recognized as the moment-generating function of a Poisson random variable with parameter λ₁ + λ₂. Therefore, the sum of two independent Poisson random variables with parameters λ₁ and λ₂ is a Poisson random variable with parameter equal to the sum λ₁ + λ₂.

   EXERCISES FOR SECTION 5-6

Problem available in WileyPLUS at instructor's discretion.

Go Tutorial Tutoring problem available in WileyPLUS at instructor's discretion

5-91. A random variable X has the discrete uniform distribution

(a) Show that the moment-generating function is

(b) Use M_X(t) to find the mean and variance of X.

5-92. A random variable X has the Poisson distribution

(a) Show that the moment-generating function is

(b) Use M_X(t) to find the mean and variance of the Poisson random variable.

5-93. The geometric random variable X has probability distribution

f(x) = (1 − p)^{x − 1}p, x = 1,2,. . .

Show that the moment-generating function is

Use M_X(t) to find the mean and variance of X.

5-94. The chi-squared random variable with k degrees of freedom has moment-generating function M_X(t) = (1 − 2t) − k/2. Suppose that X₁ and X₂ are independent chi-squared random variables with k₁ and k₂ degrees of freedom, respectively. What is the distribution of Y = X₁ + X₂?

5-95. A continuous random variable X has the following probability distribution:

(a) Find the moment-generating function for X.

(b) Find the mean and variance of X.

5-96. The continuous uniform random variable X has density function

(a) Show that the moment-generating function is

(b) Use M_X(t) to find the mean and variance of X.

5-97. A random variable X has the exponential distribution

Show that the moment-generating function of X is

(b) Find the mean and variance of X.

5-98. A random variable X has the gamma distribution

(a) Show that the moment-generating function of X is

(b) Find the mean and variance of X.

5-99. Let X₁, X₂,..., X_r be independent exponential random variables with parameter λ.

(a) Find the moment-generating function of Y = X₁ + X₂ + . . . + X_r.

(b) What is the distribution of the random variable Y?

5-100. Suppose that X_i has a normal distribution with mean μ_i and variance , i = 1, 2. Let X₁ and X₂ be independent.

(a) Find the moment-generating function of Y = X₁ + X₁.

(b) What is the distribution of the random variable Y?

   Supplemental Exercises

5-101. Show that the following function satisfies the properties of a joint probability mass function:

Determine the following:

(a) P(X < 0.5, Y < 1.5)

(b) P(X ≤ 1)

(c) P(X < 1.5)

(d) P(X > 0.5, Y < 1.5)

(e) E(X), E(Y), V(X), V(Y).

(f) Marginal probability distribution of the random variable X

(g) Conditional probability distribution of Y given that X = 1

(h) E(Y|X = 1)

(i) Are X and Y independent? Why or why not?

(j) Correlation between X and Y.

5-102. The percentage of people given an antirheumatoid medication who suffer severe, moderate, or minor side effects are 10, 20, and 70%, respectively. Assume that people react independently and that 20 people are given the medication.

Determine the following:

(a) Probability that 2, 4, and 14 people will suffer severe, moderate, or minor side effects, respectively

(b) Probability that no one will suffer severe side effects

(c) Mean and variance of the number of people who will suffer severe side effects

(d) Conditional probability distribution of the number of people who suffer severe side effects given that 19 suffer minor side effects

(e) Conditional mean of the number of people who suffer severe side effects given that 19 suffer minor side effects

5-103. The backoff torque required to remove bolts in a steel plate is rated as high, moderate, or low. Historically, the probability of a high, moderate, or low rating is 0.6, 0.3, or 0.1, respectively. Suppose that 20 bolts are evaluated and that the torque ratings are independent.

(a) What is the probability that 12, 6, and 2 bolts are rated as high, moderate, and low, respectively?

(b) What is the marginal distribution of the number of bolts rated low?

(c) What is the expected number of bolts rated low?

(d) What is the probability that the number of bolts rated low is more than two?

(e) What is the conditional distribution of the number of bolts rated low given that 16 bolts are rated high?

(f) What is the conditional expected number of bolts rated low given that 16 bolts are rated high?

(g) Are the numbers of bolts rated high and low independent random variables?

5-104. To evaluate the technical support from a computer manufacturer, the number of rings before a call is answered by a service representative is tracked. Historically, 70% of the calls are answered in two rings or less, 25% are answered in three or four rings, and the remaining calls require five rings or more. Suppose that you call this manufacturer 10 times and assume that the calls are independent.

(a) What is the probability that eight calls are answered in two rings or less, one call is answered in three or four rings, and one call requires five rings or more?

(b) What is the probability that all 10 calls are answered in four rings or less?

(c) What is the expected number of calls answered in four rings or less?

(d) What is the conditional distribution of the number of calls requiring five rings or more given that eight calls are answered in two rings or less?

(e) What is the conditional expected number of calls requiring five rings or more given that eight calls are answered in two rings or less?

(f) Is the number of calls answered in two rings or less and the number of calls requiring five rings or more independent random variables?

5-105. Determine the value of c such that the function f(x, y) = cx²y for 0 < x < 3 and 0 < y < 2 satisfies the properties of a joint probability density function.

Determine the following:

(a) P(X < 1, Y < 1)

(b) P(X < 2.5)

(c) P(1 < Y < 2.5)

(d) P(X > 2.1 < Y < 1.5)

(e) E(X)

(f) E(Y)

(g) Marginal probability distribution of the random variable X

(h) Conditional probability distribution of Y given that X = 1

(i) Conditional probability distribution of X given that Y = 1

5-106. The joint distribution of the continuous random variables X, Y, and Z is constant over the region x² + y² ≤ 1, 0 < z < 4.

Determine the following:

(a) P(X² + Y² ≤ 0.5)

(b) P(X² + Y² ≤ 0.5, Z < 2)

(c) Joint conditional probability density function of X and Y given that Z = 1

(d) Marginal probability density function of X

(e) Conditional mean of Z given that X = 0 and Y = 0

(f) Conditional mean of Z given that X = x and Y = y

5-107. Suppose that X and Y are independent, continuous uniform random variables for 0 < x < 1 and 0 < y < 1. Use the joint probability density function to determine the probability that |X − Y| <0.5.

5-108. The lifetimes of six major components in a copier are independent exponential random variables with means of 8000, 10,000, 10,000, 20,000, 20,000, and 25,000 hours, respectively.

(a) What is the probability that the lifetimes of all the components exceed 5000 hours?

(b) What is the probability that at least one component's lifetime exceeds 25,000 hours?

5-109. Contamination problems in semiconductor manufacturing can result in a functional defect, a minor defect, or no defect in the final product. Suppose that 20%, 50%, and 30% of the contamination problems result in functional, minor, and no defects, respectively. Assume that the defects of 10 contamination problems are independent.

(a) What is the probability that the 10 contamination problems result in two functional defects and five minor defects?

(b) What is the distribution of the number of contamination problems that result in no defects?

(c) What is the expected number of contamination problems that result in no defects?

5-110. The weight of adobe bricks for construction is normally distributed with a mean of 3 pounds and a standard deviation of 0.25 pound. Assume that the weights of the bricks are independent and that a random sample of 25 bricks is chosen.

(a) What is the probability that the mean weight of the sample is less than 2.95 pounds?

(b) What value will the mean weight exceed with probability 0.99?

5-111. The length and width of panels used for interior doors (in inches) are denoted as X and Y, respectively. Suppose that X and Y are independent, continuous uniform random variables for 17.75 < x < 18.25 and 4.75 < y < 5.25, respectively.

(a) By integrating the joint probability density function over the appropriate region, determine the probability that the area of a panel exceeds 90 square inches.

(b) What is the probability that the perimeter of a panel exceeds 46 inches?

5-112. The weight of a small candy is normally distributed with a mean of 0.1 ounce and a standard deviation of 0.01 ounce. Suppose that 16 candies are placed in a package and that the weights are independent.

(a) What are the mean and variance of the package's net weight?

(b) What is the probability that the net weight of a package is less than 1.6 ounces?

(c) If 17 candies are placed in each package, what is the probability that the net weight of a package is less than 1.6 ounces?

5-113. The time for an automated system in a warehouse to locate a part is normally distributed with a mean of 45 seconds and a standard deviation of 30 seconds. Suppose that independent requests are made for 10 parts.

(a) What is the probability that the average time to locate 10 parts exceeds 60 seconds?

(b) What is the probability that the total time to locate 10 parts exceeds 600 seconds?

5-114. A mechanical assembly used in an automobile engine contains four major components. The weights of the components are independent and normally distributed with the following means and standard deviations (in ounces):

(a) What is the probability that the weight of an assembly exceeds 29.5 ounces?

(b) What is the probability that the mean weight of eight independent assemblies exceeds 29 ounces?

5-115. Suppose that X and Y have a bivariate normal distribution with σ_X = 4, σ_Y = 1, /, μ_Y = 4, and ρ = −0.2. Draw a rough contour plot of the joint probability density function.

determine E(X), E(Y) V(X), V(Y), and ρ by reorganizing the parameters in the joint probability density function.

5-117. The permeability of a membrane used as a moisture barrier in a biological application depends on the thickness of two integrated layers. The layers are normally distributed with means of 0.5 and 1 millimeters, respectively. The standard deviations of layer thickness are 0.1 and 0.2 millimeters, respectively. The correlation between layers is 0.7.

(a) Determine the mean and variance of the total thickness of the two layers.

(b) What is the probability that the total thickness is less than 1 millimeter?

(c) Let X₁ and X₂ denote the thickness of layers 1 and 2, respectively. A measure of performance of the membrane is a function of 2X₁ + 3X₂ of the thickness. Determine the mean and variance of this performance measure.

5-118. The permeability of a membrane used as a moisture barrier in a biological application depends on the thickness of three integrated layers. Layers 1, 2, and 3 are normally distributed with means of 0.5, 1, and 1.5 millimeters, respectively. The standard deviations of layer thickness are 0.1, 0.2, and 0.3, respectively. Also, the correlation between layers 1 and 2 is 0.7, between layers 2 and 3 is 0.5, and between layers 1 and 3 is 0.3.

(a) Determine the mean and variance of the total thickness of the three layers.

(b) What is the probability that the total thickness is less than 1.5 millimeters?

5-119. A small company is to decide what investments to use for cash generated from operations. Each investment has a mean and standard deviation associated with the percentage gain. The first security has a mean percentage gain of 5% with a standard deviation of 2%, and the second security provides the same mean of 5% with a standard deviation of 4%. The securities have a correlation of –0.5, so there is a negative correlation between the percentage returns. If the company invests two million dollars with half in each security, what are the mean and standard deviation of the percentage return? Compare the standard deviation of this strategy to one that invests the two million dollars into the first security only.

5-120. An order of 15 printers contains 4 with a graphics-enhancement feature, 5 with extra memory, and 6 with both features. Four printers are selected at random, without replacement, from this set. Let the random variables X, Y, and Z denote the number of printers in the sample with graphics enhancement only, extra memory only, and both, respectively.

(a) Describe the range of the joint probability distribution of X, Y, and Z.

(b) Is the probability distribution of X, Y, and Z multinomial? Why or why not?

(c) Determine the conditional probability distribution of X given that Y = 2.

Determine the following:

(d) P(X = 1, Y = 2, Z = 1)

(e) P(X = 1, Y = 1)

(f) E(X) and V(X)

(g) P(X = 1, Y = 2|Z = 1)

(h) P(X = 2|Y = 2)

(i) Conditional probability distribution of X given that Y = 0 and Z = 3.

5-121. A marketing company performed a risk analysis for a manufacturer of synthetic fibers and concluded that new competitors present no risk 13% of the time (due mostly to the diversity of fibers manufactured), moderate risk 72% of the time (some overlapping of products), and very high risk (competitor manufactures the exact same products) 15% of the time. It is known that 12 international companies are planning to open new facilities for the manufacture of synthetic fibers within the next three years. Assume that the companies are independent. Let X, Y, and Z denote the number of new competitors that will pose no, moderate, and very high risk for the interested company, respectively.

Determine the following:

(a) Range of the joint probability distribution of X, Y, and Z

(b) P(X = 1, Y = 3, Z = 1)

(c) P(Z ≤ 2)

(d) P(Z = 2|Y = 1, X = 10)

(e) P(Z ≤ 1|X = 10)

(f) P(Y ≤ 1, Z ≤ 1|X = 10)

(g) E(Z|X = 10)

5-122. Suppose X has a lognormal distribution with parameters θ and ω. Determine the probability density function and the parameters values for Y = X^γ for a constant γ > 0. What is the name of this distribution?

5-123. The power in a DC circuit is P = I²/R where I and R denote the current and resistance, respectively. Suppose that I is normally distributed with mean of 200 mA and standard deviation 0.2 mA and R is a constant. Determine the probability density function of power.

5-124. The intensity (mW/mm²) of a laser beam on a surface theoretically follows a bivariate normal distribution with maximum intensity at the center, equal variance σ in the x and y directions, and zero covariance. There are several definitions for the width of the beam. One definition is the diameter at which the intensity is 50% of its peak. Suppose that the beam width is 1.6 mm under this definition. Determine σ. Also determine the beam width when it is defined as the diameter where the intensity equals 1/e² of the peak.

5-125. Use moment generating functions to determine the normalized power [E(X⁴)]^¼ from a cycling power meter when X has a normal distribution with mean 200 and standard deviation 20 Watts.