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Applied Statistics and Probability for Engineers

Douglas C. Montgomery

Chapter 5

Joint Probability Distributions - all with Video Answers

Educators

Section 1

Two or More Random Variables

12:12

Problem 1

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?

Amany Waheeb
Amany Waheeb
Numerade Educator
10:21

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

Amany Waheeb
Amany Waheeb
Numerade Educator
08:06

Problem 3

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?

Amany Waheeb
Amany Waheeb
Numerade Educator
01:26

Problem 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 $Y$

Amany Waheeb
Amany Waheeb
Numerade Educator
19:32

Problem 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) $f_{X Y}(x, y)$
(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?

Amany Waheeb
Amany Waheeb
Numerade Educator
10:57

Problem 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) $f_{X Y}(x, y)$
(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?

Amany Waheeb
Amany Waheeb
Numerade Educator
01:11

Problem 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) $f_{X Y}(x, y)$
(b) $f_{X}(x)$
(c) $E(X)$
(d) $f_{Y \mid 2}(y)$
(e) $E(Y \mid X=2)$
(f) $V(Y \mid X=2)$
(g) Are $X$ and $Y$ independent?

Victor Salazar
Victor Salazar
Numerade Educator
07:19

Problem 8

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 $Z=2$

Amany Waheeb
Amany Waheeb
Numerade Educator
10:42

Problem 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) $f_{X Y}(x, y)$
(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?

Amany Waheeb
Amany Waheeb
Numerade Educator
03:35

Problem 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) $P(X<6 \mid Y=0)$
(e) $E(X \mid Y=0)$

Amany Waheeb
Amany Waheeb
Numerade Educator
05:23

Problem 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) $f_{X Y Z}(x, y, z)$
(b) Conditional probability mass function for $X$ and $Y$ given $Z=0$
(c) $P(X<6, Y<6 \mid Z=0)$
(d) $E(X \mid Y=0, Z=0)$

Amany Waheeb
Amany Waheeb
Numerade Educator
06:00

Problem 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:
(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 $Y=2$

Amany Waheeb
Amany Waheeb
Numerade Educator
09:39

Problem 13

Determine the value of $c$ such that the function $f(x, y)=c x y$ 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) $E(Y \mid X)=1.5)$
(j) $P(Y<2 \mid X=1.5)$
(k) Conditional probability distribution of $X$ given that $Y=2$

Amany Waheeb
Amany Waheeb
Numerade Educator
12:30

Problem 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) $E(Y \mid X=1)$
(j) $P(Y>2 \mid X=1)$
(k) Conditional probability distribution of $X$ given that $Y=2$

Amany Waheeb
Amany Waheeb
Numerade Educator
03:14

Problem 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) $E(Y \mid X=1)$
(j) $P(Y>2 \mid X=1)$
(k) Conditional probability distribution of $X$ given $Y=2$

Victor Salazar
Victor Salazar
Numerade Educator
02:41

Problem 16

Determine the value of $c$ that makes the function $f(x, y)=c e^{-2 x-3 y}$ 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) $E(Y \mid X=1)$
(j) Conditional probability distribution of $X$ given $Y=2$

Victor Salazar
Victor Salazar
Numerade Educator
02:20

Problem 17

Determine the value of $c$ that makes the function $f(x, y)=c e^{-2 x-3 y},$ 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) $E(Y \mid X=1)$
(j) $P(Y<2 \mid X=1)$
(k) Conditional probability distribution of $X$ given $Y=2$

Victor Salazar
Victor Salazar
Numerade Educator
13:09

Problem 18

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) $f_{Y}(y)$

Mengchun Cai
Mengchun Cai
Numerade Educator
02:10

Problem 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, $f_{X Y}(x, y)=c$ for $x$ and $y$ in the region. Determine the value for $c$ such that $f_{X Y}(x, y)$ 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 $X=1$
$(\mathrm{g}) E(Y \mid X=1)$
(h) $P(Y<0.5 \mid X=1)$

Victor Salazar
Victor Salazar
Numerade Educator
03:39

Problem 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?

Amany Waheeb
Amany Waheeb
Numerade Educator
04:36

Problem 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?

Amany Waheeb
Amany Waheeb
Numerade Educator
02:55

Problem 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?

Amany Waheeb
Amany Waheeb
Numerade Educator
01:27

Problem 23

Suppose that the random variables $X, Y,$ and $Z$ have the joint probability density function $f(x, y, z)=8 x y z$ 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) $P(X<0.5 \mid Y=0.5)$
(g) $P(X<0.5, Y<0.5 \mid Z=0.8)$
(h) Conditional probability distribution of $X$ given that $Y=0.5$ and $Z=0.8$
(i) $P(X<0.5 \mid Y=0.5, Z=0.8)$

Victor Salazar
Victor Salazar
Numerade Educator
01:18

Problem 24

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 $0<z<4 .$ Determine the constant $c$ so that $f_{X Y Z}(x, y, z)$ is a probability density function. Determine the following:
(a) $P\left(X^{2}+Y^{2}<2\right)$
(b) $P(Z<2)$
(c) $E(X)$
(d) $P(X<1 \mid Y=1)$
(e) $P\left(X^{2}+Y^{2}<1 \mid Z=1\right)$
(f) Conditional probability distribution of $Z$ given that $X=1$ and $Y=1$

Victor Salazar
Victor Salazar
Numerade Educator
02:14

Problem 25

Determine the value of $c$ that makes $f_{X Y Z}(x, y, z)=c$ 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 $Y=0.5$

Victor Salazar
Victor Salazar
Numerade Educator
03:08

Problem 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?

Amany Waheeb
Amany Waheeb
Numerade Educator
02:45

Problem 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?

Amany Waheeb
Amany Waheeb
Numerade Educator
01:33

Problem 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?

Victor Salazar
Victor Salazar
Numerade Educator
03:09

Problem 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 $f_{Y \mid x}(y)=\exp [-(y-x)], x<y .$ Answer or determine the following:
(a) That $f_{Y \mid x}(y)$ 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 $f_{X Y}(x, y)$.
(d) Conditional probability density function of $X$ given $Y=y$.
(e) $P(Y<2 \mid X=1)$
(f) $E(Y \mid X=1)$
(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?

Victor Salazar
Victor Salazar
Numerade Educator
03:20

Problem 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 $\geq 75 .$ 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, $P(Y=50 \mid X=75)$ $=0.0717$. 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) $P(Y \leq 50 \mid X=50)$
(b) $P(X=0, Y=75)$
(c) $E(Y \mid X=50)$
(d) $f_{Y}(y)$
(e) $f_{X Y}(x, y)$
(f) Are $X$ and $Y$ independent?

Victor Salazar
Victor Salazar
Numerade Educator
00:48

Problem 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.

Nick Johnson
Nick Johnson
Numerade Educator
00:53

Problem 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 $1.6 x$ 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 $Y$

Victor Salazar
Victor Salazar
Numerade Educator