Chapter 5

Show that the following function satisfies the properties of a joint probability mass function. $$ \begin{array}{ccc} \hline x & y & f_{X Y}(x, y) \\ \hline 1.0 & 1 & 1 / 4 \\ 1.5 & 2 & 1 / 8 \\ 1.5 & 3 & 1 / 4 \\ 2.5 & 4 & 1 / 4 \\ 3.0 & 5 & 1 / 8 \end{array} $$ 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) \(E(Y \mid X=1.5)\) (j) Are \(X\) and \(Y\) independent?

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) \(E(Y \mid X=1)\) (j) Are \(X\) and \(Y\) independent?

Show that the following function satisfies the properties of a joint probability mass function. $$ \begin{array}{|c|c|c|} \hline x & y & f_{X Y}(x, y) \\ \hline-1.0 & -2 & 1 / 8 \\ -0.5 & -1 & 1 / 4 \\ \hline 0.5 & 1 & 1 / 2 \\ \hline 1.0 & 2 & 1 / 8 \\ \hline \end{array} $$ 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) \(E(X \mid y=1)\) (j) Are \(X\) and \(Y\) independent?

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 \(Y\)

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) \(f_{X Y}(x, y)\) (b) \(f_{X}(x)\) (c) \(E(X)\) (d) \(f_{Y \mid 1}(y)\) (e) \(E(Y \mid X=1)\) (f) Are \(X\) and \(Y\) independent?

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) \(f_{X Y}(x, y)\) (b) \(f_{X}(x)\) (c) \(E(X)\) (d) \(f_{Y \mid 3}(y)\) (e) \(E(Y \mid X=3)\) (f) \(V(Y \mid X=3)\) (g) Are \(X\) and \(Y\) independent?

Suppose that the random variables \(X, Y,\) and \(Z\) have the following joint probability distribution. $$ \begin{array}{|cc|c|c|} \hline x & y & z & f(x, y, z) \\ \hline 1 & 1 & 1 & 0.05 \\ \hline 1 & 1 & 2 & 0.10 \\ \hline 1 & 2 & 1 & 0.15 \\ \hline 1 & 2 & 2 & 0.20 \\ \hline 2 & 1 & 1 & 0.20 \\ \hline 2 & 1 & 2 & 0.15 \\ \hline 2 & 2 & 1 & 0.10 \\ \hline 2 & 2 & 2 & 0.05 \\ \hline \end{array} $$ Determine the following: (a) \(P(X=2)\) (b) \(P(X=1, Y=2)\) (c) \(P(Z<1.5)\) (d) \(P(X=1 \quad\) or \(\quad Z=2)\) (e) \(E(X)\) (f) \(P(X=1 \mid Y=1)\) (g) \(P(X=1, Y=1 \mid Z=2)\) (h) \(P(X=1 \mid Y=1, Z=2)\) (i) Conditional probability distribution of \(X\) given that \(Y=1\) and \(Z=2\)

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) \(f_{X Y}(x, y)\) (b) \(f_{X}(x)\) (c) \(E(X)\) (d) \(f_{Y \mid 3}(y)\) (e) \(E(Y \mid X=3)\) (f) \(V(Y \mid X=3)\) (g) Are \(X\) and \(Y\) independent?

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: (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 \(Y=2\)

Determine the value of \(c\) such that the function \(f(x, y)=c x y\) for \(01.8,1

Determine the value of \(c\) that makes the function \(f(x, y)=c(x+y)\) a joint probability density function over the range \(01)\) (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) \(E(Y \mid X=1)\) (j) \(P(Y>2 \mid X=1)\) (k) Conditional probability distribution of \(X\) given that \(Y=2\)

Determine the value of \(c\) that makes the function \(f(x, y)=c(x+y)\) a joint probability density function over the range \(01)\) (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) \(E(Y \mid X=1)\) (j) \(P(Y>2 \mid X=1)\) (k) Conditional probability distribution of \(X\) given \(Y=2\)

The conditional probability distribution of \(Y\) given \(X=x\) is \(f_{Y \mid x}(y)=x e^{-x y}\) for \(y>0,\) and the marginal probability distribution of \(X\) is a continuous uniform distribution over 0 to \(10 .\) (a) Graph \(f_{Y \mid X}(y)=x e^{-x y}\) for \(y>0\) for several values of \(x\). Determine: (b) \(P(Y<2 \mid X=2)\) (c) \(E(Y \mid X=2)\) (d) \(E(Y \mid X=x)\) (e) \(f_{X Y}(x, y)\) (f) \(f_{Y}(y)\)

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

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?

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?

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?

Suppose that the random variables \(X, Y,\) and \(Z\) have the joint probability density function \(f_{X Y Z}(x, y, z)=c\) over the cylinder \(x^{2}+y^{2}<4\) and \(0

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?

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?

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?

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

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.

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 \(1.6 x\) when \(X=x\) and standard deviation of \(10 .\) Determine the following: (a) Conditional probability density function \(f_{Y \mid 73}(y)\) of \(Y\) given \(X=73\) (b) \(P(Y<115 \mid X=73)\) (c) \(E(Y \mid X=73)\) (d) Recognize the distribution \(f_{X Y}(x, y)\) and identify the mean and variance of \(Y\) and the correlation between \(X\) and \(Y\)

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 \(Y\).

Determine the value for \(c\) and the covariance and correlation for the joint probability density function \(f_{X Y}(x, y)=c x y\) over the range \(0

Determine the covariance and correlation for the joint probability density function \(f_{X Y}(x, y)=e^{-x-y}\) over the range \(0

The joint probability distribution is $$ \begin{array}{lcccc} x & -1 & 0 & 0 & 1 \\ y & 0 & -1 & 1 & 0 \\ f_{X Y}(x, y) & 1 / 4 & 1 / 4 & 1 / 4 & 1 / 4 \end{array} $$ Show that the correlation between \(X\) and \(Y\) is zero but \(X\) and \(Y\) are not independent.

Suppose that \(X\) and \(Y\) are independent continuous random variables. Show that \(\sigma_{X Y}=0 .\)

Suppose that the correlation between \(X\) and \(Y\) is \(\rho\). For constants \(a, b, c,\) and \(d,\) what is the correlation between the random variables \(U=a X+b\) and \(V=c Y+d ?\)

Test results from an electronic circuit board indicate that \(50 \%\) of board failures are caused by assembly defects, \(30 \%\) by electrical components, and \(20 \%\) by mechanical defects. Suppose that 10 boards fail independently. Let the random variables \(X, Y,\) and \(Z\) denote the number of assembly, electrical, and mechanical defects among the 10 boards. Calculate the following: (a) \(P(X=5, Y=3, Z=2)\) (b) \(P(X=8)\) (c) \(P(X=8 \mid Y=1)\) (d) \(P(X \geq 8 \mid Y=1)\) (e) \(P(X=7, Y=1 \mid Z=2)\)

Based on the number of voids, a ferrite slab is classified as either high, medium, or low. Historically, \(5 \%\) of the slabs are classified as high, \(85 \%\) as medium, and \(10 \%\) as low. A sample of 20 slabs is selected for testing. Let \(X, Y,\) and \(Z\) denote the number of slabs that are independently classified as high, medium, and low, respectively. (a) What are the name and the values of the parameters of the joint probability distribution of \(X, Y,\) and \(Z ?\) (b) What is the range of the joint probability distribution of \(X\), \(Y,\) and \(Z ?\) (c) What are the name and the values of the parameters of the marginal probability distribution of \(X ?\) (d) Determine \(E(X)\) and \(V(X)\). Determine the following: (e) \(P(X=1, Y=17, Z=3)\) (f) \(P(X \leq 1, Y=17, Z=3)\) (g) \(P(X \leq 1)\) (h) \(E(Y)\) (i) \(P(X=2, Z=3 \mid Y=17)\) (j) \(P(X=2 \mid Y=17)\) (k) \(E(X \mid Y=17)\)

A Web site uses ads to route visitors to one of four landing pages. The probabilities for each landing page are equal. Consider 20 independent visitors and let the random variables \(W, X, Y,\) and \(Z\) denote the number of visitors routed to each page. Calculate the following: (a) \(P(W=5, X=5, Y=5, Z=5)\) (b) \(P(W=5, X=5, Y=5, Z=5)\) (c) \(P(W=7, X=7, Y=6 \mid Z=3)\) (d) \(P(W=7, X=7, Y=3 \mid Z=3)\) (e) \(P(W \leq 2)\) (f) \(E(W)\) (g) \(P(W=5, X=5)\) (h) \(P(W=5 \mid X=5)\)

Four electronic ovens that were dropped during shipment are inspected and classified as containing either a major, a minor, or no defect. In the past, \(60 \%\) of dropped ovens had a major defect, \(30 \%\) had a minor defect, and \(10 \%\) had no defect. Assume that the defects on the four ovens occur independently. (a) Is the probability distribution of the count of ovens in each category multinomial? Why or why not? (b) What is the probability that, of the four dropped ovens, two have a major defect and two have a minor defect? (c) What is the probability that no oven has a defect? Determine the following: (d) Joint probability mass function of the number of ovens with a major defect and the number with a minor defect (e) Expected number of ovens with a major defect (f) Expected number of ovens with a minor defect (g) Conditional probability that two ovens have major defects given that two ovens have minor defects (h) Conditional probability that three ovens have major defects given that two ovens have minor defects (i) Conditional probability distribution of the number of ovens with major defects given that two ovens have minor defects (j) Conditional mean of the number of ovens with major defects given that two ovens have minor defects.

Let \(X\) and \(Y\) represent the concentration and viscosity of a chemical product. Suppose that \(X\) and \(Y\) have a bivariate normal distribution with \(\sigma_{X}=4, \sigma_{Y}=1, \mu_{X}=2\) and \(\mu_{y}=1\). Draw a rough contour plot of the joint probability density function for each of the following values of \(\rho\) : (a) \(\rho=0\) (b) \(\rho=0.8\) (c) \(\rho=-0.8\)

Suppose that \(X\) and \(Y\) have a bivariate normal distribution with \(\sigma_{X}=0.04, \sigma_{Y}=0.08, \mu_{X}=3.00, \mu_{Y}=7.70,\) and \(\rho=0 .\) Determine the following: (a) \(P(2.95

In an acid-base titration, a base or acid is gradually added to the other until they have completely neutralized each other. Let \(X\) and \(Y\) denote the milliliters of acid and base needed for equivalence, respectively. Assume that \(X\) and \(Y\) have a bivariate normal distribution with \(\sigma_{X}=5 \mathrm{~mL}, \sigma_{Y}=2 \mathrm{~mL}\), \(\mu_{X}=120 \mathrm{~mL}, \mu_{Y}=100 \mathrm{~mL},\) and \(\rho=0.6\) Determine the following: (a) Covariance between \(X\) and \(Y\) (b) Marginal probability distribution of \(X\) (c) \(P(X<116)\) (d) Conditional probability distribution of \(X\) given that \(Y=102\) (e) \(P(X<116 \mid Y=102)\)

In the manufacture of electroluminescent lamps, several different layers of ink are deposited onto a plastic substrate. The thickness of these layers is critical if specifications regarding the final color and intensity of light are to be met. Let \(X\) and \(Y\) denote the thickness of two different layers of ink. It is known that \(X\) is normally distributed with a mean of 0.1 millimeter and a standard deviation of 0.00031 millimeter, and \(Y\) is normally distributed with a mean of 0.23 millimeter and a standard deviation of 0.00017 millimeter. The value of \(\rho\) for these variables is equal to \(0 .\) Specifications call for a lamp to have a thickness of the ink corresponding to \(X\) in the range of 0.099535 to 0.100465 millimeter and \(Y\) in the range of 0.22966 to 0.23034 millimeter. What is the probability that a randomly selected lamp will conform to specifications?

Patients given drug therapy either improve, remain the same, or degrade with probabilities \(0.5,0.4,0.1,\) respectively. Suppose that 20 patients (assumed to be independent) are given the therapy. Let \(X_{1}, X_{2},\) and \(X_{3}\) denote the number of patients who improved, stayed the same, or became degraded. Determine the following. (a) Are \(X_{1}, X_{2}, X_{3}\) independent? (b) \(P\left(X_{1}=10\right)\) (c) \(P\left(X_{1}=10, X_{2}=8, X_{3}=2\right)\) (d) \(P\left(X_{1}=5 \mid X_{2}=12\right)\) (e) \(E\left(X_{1}\right)\)

Suppose that \(X\) has a standard normal distribution. Let the conditional distribution of \(Y\) given \(X=x\) be normally distributed with mean \(E(Y \mid x)=2 x\) and variance \(V(Y \mid x)=2 x\). Determine the following. (a) Are \(X\) and \(Y\) independent? (b) \(P(Y<3 \mid X=3)\) (c) \(E(Y \mid X=3)\) (d) \(f_{X Y}(x, y)\) (e) Recognize the distribution \(f_{X Y}(x, y)\) and identify the mean and variance of \(Y\) and the correlation between \(X\) and \(Y\).

Suppose that \(X\) and \(Y\) have a bivariate normal distribution with joint probability density function \(f_{X Y}\left(x, y ; \sigma_{X}, \sigma_{Y}, \mu_{X}, \mu_{Y}, \rho\right)\) (a) Show that the conditional distribution of \(Y\) given that \(X=x\) is normal. (b) \(\quad\) Determine \(E(Y \mid X=x)\). (c) \(\quad\) Determine \(V(Y \mid X=x)\).

If \(X\) and \(Y\) have a bivariate normal distribution with \(\rho=0,\) show that \(X\) and \(Y\) are independent.

Show that the probability density function \(f_{X Y}\) \(\left(x, y ; \sigma_{X}, \sigma_{Y}, \mu_{X}, \mu_{Y}, \rho\right)\) of a bivariate normal distribution integrates to 1. [Hint: Complete the square in the exponent and use the fact that the integral of a normal probability density function for a single variable is \(1 .]\)

If \(X\) and \(Y\) have a bivariate normal distribution with joint probability density \(f_{X Y}\left(x, y ; \sigma_{X}, \sigma_{Y}, \mu_{X}, \mu_{Y}, \rho\right),\) show that the marginal probability distribution of \(X\) is normal with mean \(\mu_{x}\) and standard deviation \(\sigma_{x} .\) [Hint: Complete the square in the exponent and use the fact that the integral of a normal probability density function for a single variable is \(1 .]\)

\(X\) and \(Y\) are independent, normal random variables with \(E(X)=0, V(X)=4, E(Y)=10,\) and \(V(Y)=9\) Determine the following: (a) \(E(2 X+3 Y)\) (b) \(V(2 X+3 Y)\) (c) \(P(2 X+3 Y<30)\) (d) \(P(2 X+3 Y<40)\)

\(X\) and \(Y\) are independent, normal random variables with \(E(X)=2, V(X)=5, E(Y)=6,\) and \(V(Y)=8 .\) Determine the following: (a) \(E(3 X+2 Y)\) (b) \(V(3 X+2 Y)\) (c) \(P(3 X+2 Y<18)\) (d) \(P(3 X+2 Y<28)\)

Suppose that the random variable \(X\) represents the length of a punched part in centimeters. Let \(Y\) be the length of the part in millimeters. If \(E(X)=5\) and \(V(X)=0.25,\) what are the mean and variance of \(Y ?\)

A plastic casing for a magnetic disk is composed of two halves. The thickness of each half is normally distributed with a mean of 2 millimeters, and a standard deviation of 0.1 millimeter and the halves are independent. (a) Determine the mean and standard deviation of the total thickness of the two halves. (b) What is the probability that the total thickness exceeds 4.3 millimeters?

Making handcrafted pottery generally takes two major steps: wheel throwing and firing. The time of wheel throwing and the time of firing are normally distributed random variables with means of 40 minutes and 60 minutes and standard deviations of 2 minutes and 3 minutes, respectively. (a) What is the probability that a piece of pottery will be finished within 95 minutes? (b) What is the probability that it will take longer than 110 minutes?

In the manufacture of electroluminescent lamps, several different layers of ink are deposited onto a plastic substrate. The thickness of these layers is critical if specifications regarding the final color and intensity of light are to be met. Let \(X\) and \(Y\) denote the thickness of two different layers of ink. It is known that \(X\) is normally distributed with a mean of \(0.1 \mathrm{~mm}\) and a standard deviation of \(0.00031 \mathrm{~mm},\) and \(Y\) is also normally distributed with a mean of \(0.23 \mathrm{~mm}\) and a standard deviation of \(0.00017 \mathrm{~mm}\). Assume that these variables are independent. (a) If a particular lamp is made up of these two inks only, what is the probability that the total ink thickness is less than \(0.2337 \mathrm{~mm} ?\) (b) A lamp with a total ink thickness exceeding \(0.2405 \mathrm{~mm}\) lacks the uniformity of color that the customer demands. Find the probability that a randomly selected lamp fails to meet customer specifications.