Statistics; Probability, Inference, and Decision

William Lee Hays; Robert L. Winkler

Chapter 3 Probability Distributions - all with Video Answers

Educators

Chapter Questions

Problem 1

The random variable $X$ has the following probability distribution:
$$
\begin{array}{rc}
\hline x & P(X=x) \\
\hline-1 & .2 \\
0 & .3 \\
3 & .2 \\
4 & .2 \\
6 & .1 \\
\hline
\end{array}
$$
(a) graph the PMF and the CDF of $X$
(b) explain the relationship between the PMF and the CDF
(c) find $P(X \geq 2.5)$
(d) find $P(-1<X<4)$
(e) find $P(-1 \leq X \leq 4)$
(f) find $P(X<-3)$
(g) find $P(X=1)$.

Sonam Khatri

Numerade Educator

00:59

Problem 2

The random variable $X$ has the probability distribution given by the rule
$$
P(X=a)= \begin{cases}1 / k & \text { for } a=1,2,3,4,5, \\ 0 & \text { elsewhere } .\end{cases}
$$
(a) is $X$ discrete or continuous?
(b) find $k$
(c) graph the distribution of $X$ in two ways
(d) find the distribution of $Y=(X-3)^2$ and graph it.

Sriparna Bhattacharjee

Numerade Educator

Problem 3

The cumulative distribution function of $X$ is given by the rule
$$
F(x)= \begin{cases}1 & \text { if } x \geq 2, \\ \frac{1}{4} & \text { if } 1 \leq x<2, \\ 0 & \text { if } x<1 .\end{cases}
$$
(a) find the corresponding PMF
(b) find $P(1<X<2)$.

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03:49

Problem 4

The density function of $X$ is given by
$$
f(x)= \begin{cases}k x(1-x) & \text { for } 0<x<1, \\ 0 & \text { elsewhere. }\end{cases}
$$
(a) find $k$ and graph the density function
(b) find $P\left(\frac{1}{4}<X<\frac{1}{2}\right)$
(c) find $P\left(-\frac{1}{2} \leq X \leq \frac{1}{4}\right)$
(d) find the CDF and graph it.

Sheryl Ezze

Numerade Educator

03:19

Problem 5

The CDF of $X$ is given by
$$
F(x)= \begin{cases}1 & \text { for } x \geq 2 \\ x^2 / 4 & \text { for } 0 \leq x<2 \\ 0 & \text { for } x<0\end{cases}
$$
(a) find $f(x)$, the density function, and show that it satisfies the two requirements for a density function
(b) graph $f(x)$ and $F(x)$.

Sheryl Ezze

Numerade Educator

02:01

Problem 6

Suppose that the face value of a playing card is regarded as a random variable, with an ace counting as 1 and any face card (Jack, Queen, King) counting as 10. You draw one card at random from a well-shuffled deck. Construct a PMF showing the probability distribution for this random variable and find the probabilities of the following events:
(a) $P(X \leq 6)$
(d) $P(X$ is an even number $)$
(b) $P(4<X)$
(e) $P(X \neq 2 \cap X \neq 8)$.
(c) $P[(2 \leq X \leq$
7) $\cup(X=10)]$

Bailey Brooks

Numerade Educator

02:22

Problem 7

The density function of $X$ is given by
$$
\begin{aligned}
f(x) & =\frac{2(3-x)}{9} & \text { for } 0 \leq x \leq 3, \\
& =0 & \text { elsewhere. }
\end{aligned}
$$
Without using integration, show that the area under the curve is equal to one and find $P(1<X<1.5)$ and $P(X>2)$. [Hint: Use the fact that the area within a triangle is equal to $\frac{1}{2}$ of the product of the base and the height.]

Christopher Stanley

Numerade Educator

03:23

Problem 8

Suppose that the random variable $X$ represents the number of heads occurring in three independent tosses of a fair coin. Represent the distribution of $X$
(a) by a listing
(b) by a graph of the PMF
(c) by a graph of the CDF.
Do the same for $Y$, the number of heads occurring in four independent tosses of a fair coin.

Lucas Finney

Numerade Educator

01:06

Problem 9

Discuss the proposition: "All observed numerical events represent values of discrete variables, and continuous variables are only an idealization."

Trent Speier

Numerade Educator

02:00

Problem 10

Why is it necessary to deal with probability densities rather than probabilities such as $P(X=a)$ when the variable under consideration is continuous?

Gopesh Vishwakarma

Numerade Educator

05:30

Problem 11

The density function of $X$ is given by the rule
$$
f(x)= \begin{cases}x & \text { for } 0<x \leq 1 \\ 2-x & \text { for } 1<x<2 \\ 0 & \text { elsewhere }\end{cases}
$$
(a) find the CDF
(b) prove that $f(x)$ satisfies the two requirements for a density function
(c) find $P\left(\frac{1}{2}<\mathrm{X}<\frac{3}{2}\right)$.

Willis James

Numerade Educator

07:25

Problem 12

Does the function
$$
f(x)= \begin{cases}2 x / 3 & \text { for }-1 \leq x \leq 2, \\ 0 & \text { elsewhere }\end{cases}
$$
satisfy the two requirements for a density function?

Amany Waheeb

Numerade Educator

01:33

Problem 13

Suppose that a person agrees to pay you $$\$ 10$$ if you throw at least one six in four tosses of a fair die. How much would you have to pay him for this opportunity in order to make it a "fair" gamble?

Prabhakar Kumar

Numerade Educator

00:14

Problem 14

Explain what is meant by the statement, "on any single trial, we do not expect the expectation."

Ahmed Genedy

Numerade Educator

01:15

Problem 15

For the distribution of Exercise 1, find the mean, the mode, and the median. Is the distribution skewed positively or negatively?

Adriano Chikande

Numerade Educator

01:24

Problem 16

For the distribution of Exercise 2, find the mean, the mode, the median, and the mid-range and compare these measures of location, or central tendency. Is this distribution symmetric or skewed?

Charles Machakwa

Numerade Educator

08:56

Problem 17

Find the following fractiles of the distribution in Exercise 1:
(a) .25
(b) .75
(c) .01
(d) .33
(e) .90 .

Norman Atentar

Numerade Educator

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Problem 18

Explain why it is possible for a discrete random variable to have more than one median.

Shu Naito

Numerade Educator

09:35

Problem 19

For the distribution of Exercise 5, find the mean, the mode, and the median, and comment on the relative value of these measures in this example as measures of the "typical value" of the random variable.

Evelyn Cunningham

Numerade Educator

00:27

Problem 20

Can you determine the mode of a distribution by examining the CDF? Explain for both discrete and continuous distributions.

Nick Johnson

Numerade Educator

01:51

Problem 21

Suppose that $X$ represents the daily sales of a particular product, and that the probability distribution of $X$ is as follows:
$$
\begin{array}{cc}
\hline x & P(X=x) \\
\hline 7,000 & .05 \\
7,500 & .20 \\
8,000 & .35 \\
8,500 & .19 \\
9,000 & .12 \\
9,500 & .08 \\
10,000 & .01 \\
\hline
\end{array}
$$
Find the expectation and the variance of daily sales. [Hint: To find the variance, it is easiest to first find the variance of a different variable, such as $Y=(X-7000) / 1000$. ] If the net profit resulting from sales of $X$ items can be given by
$$
Z=5 X-38,000,
$$
find the expectation and the variance of net profit, $Z$.

Alexander Cheng

Numerade Educator

06:56

Problem 22

For the distribution of Exercise 1, find the variance, the standard deviation, the expected absolute deviation, and the range. Also, find these same values under the assumption that the largest value of $X$ is 12 rather than 6 . In light of your results, comment on the relative merit of the different measures of dispersion.

Samuel Goyette

Numerade Educator

04:36

Problem 23

For the distribution of Exercise 3, find
(a) $E(X)$
(b) $E\left(X^2\right)$
(c) $E(4 X+12)$
(d) $\operatorname{var}(X)$
(e) $\operatorname{var}(4 X+12)$.

Michelle Z.

Numerade Educator

01:08

Problem 24

Find the mean and variance of the random variables $X$ and $Y$ in Exercise 8 .

Yingtai Xiao

Numerade Educator

03:01

Problem 25

What do we mean when we say that the expectation of a random variable can be thought of as a center of gravity?

Maxime Rossetti

Numerade Educator

02:07

Problem 26

If a random variable has a mean of ten and a variance of zero, graph its distribution.

Lucas Finney

Numerade Educator

03:32

Problem 27

Prove that in general, $E\left(X^2\right)$ is not equal to $[E(X)]^2$.

Priyanka Sadarangani

Numerade Educator

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Problem 28

Criticize the following statement: If the mean of a distribution is 50 , and the standard deviation is 10 , then the "best bet" about any case drawn at random is 50 , and, on the average, one can expect to be in error by 10 points.

Rashmi Sinha

Numerade Educator

03:51

Problem 29

If the density function of $X$ is given by
$$
f(x)= \begin{cases}a+b x^2 & \text { for } 0 \leq x \leq 1 \\ 0 & \text { elsewhere }\end{cases}
$$
and $E(X)=\frac{2}{3}$, find $a$ and $b$.

Amany Waheeb

Numerade Educator

01:15

Problem 30

Find the following fractiles of the distribution in Exercise 5:
(a) .01
(b) .05
(c) .25
(d) .40
(e) .70
(f) .85 .

Maxime Rossetti

Numerade Educator

04:15

Problem 31

From your own main area of interest, think of two variables that would be expected to have reasonably symmetric distributions, two variables that would be expected to have positively skewed distributions, and two variables that would be expected to have negatively skewed distributions.

Kylie Patel

Numerade Educator

02:02

Problem 32

The range is the easiest to compute of all of the measures of dispersion which we discussed. In view of this, why is the range not preferred to the standard deviation, which is much more difficult to compute?

Matthias Wuest

Numerade Educator

01:43

Problem 33

What advantage does the standard deviation have over the variance as a measure of dispersion?

Sneha Ravi

Numerade Educator

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Problem 34

Suppose that the joint distribution of $X$ and $Y$ is represented by the following table:
TABLE CANT COPY
(a) graph the joint PMF of $X$ and $Y$ (you will need a three-dimensional graph)
(b) are $X$ and $Y$ independent? Explain your answer.
(c) determine the marginal distributions of $X$ and $Y$
(d) determine the conditional distribution of $X$ given that $Y=2$. What does this tell you about the conditional distribution of $X$ given any particular value of $Y$ ?
(e) find $E(X), E(Y), E(X+Y), E(X Y)$, and $E(4 X-2 Y)$
(f) find $\operatorname{cov}(X, Y)$, var $(X+Y)$, and var $(X-Y)$
(g) using the distribution obtained in (d), find $E(X \mid Y=2)$.

Amany Waheeb

Numerade Educator

Problem 35

Complete the following table, given that $X$ and $Y$ are independent.
TABLE CANT COPY

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Problem 36

Complete the following table, given that $P(X=1 \mid Y=2)=\frac{1}{3}$ and $P(Y=3 \mid X=2)=\frac{1}{2}$
TABLE CANT COPY

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02:04

Problem 37

From your major field of interest, make up two examples of variables that you might reasonably expect to be independent. Also, make up four examples of variables that should be associated to some degree, two examples involving variables with a positive relationship and two examples involving variables with a negative relationship.

Aarushi Singh

Numerade Educator

Problem 38

There is a real danger in confusing the idea of causation with that of statistical association. Why do you think these two concepts are so often confused? For example, for centuries it was thought that swampy air when breathed "caused" malaria (hence the name, literally "bad air"). Comment on what this example shows about statistical association and causation.

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02:52

Problem 39

For the distribution of Exercise 2, find the correlation coefficient of $X$ and $Y=(X-3)^2$. In this example, if we know $X$, we can easily determine $Y$ with certainty, and vice versa. If this is true, why is the correlation coefficient not equal to +1 or -1 ? What does this example illustrate about correlation and association?

Khoobchandra Agrawal

Numerade Educator

07:33

Problem 40

If $\operatorname{cov}(X, Y)=1.0$, what can we say about the relationship between $X$ and $Y$ ? If $\operatorname{cov}(Z, W)=2.0$, does this mean that the relationship between $Z$ and $W$ is stronger than the relationship between $X$ and $Y$ ? Explain.

James Kiss

Numerade Educator

02:38

Problem 41

Prove that if $X$ and $Y$ are independent, then their covariance and their correlation coefficient are equal to zero [Note: The reverse implication is not true in general.]

Heena Haldankar

Numerade Educator

01:20

Problem 42

Prove that cov $(a X, b Y)=a b \operatorname{cov}(X, Y)$.

Christopher Stanley

Numerade Educator

01:47

Problem 43

If $\operatorname{var}(X)=50$, $\operatorname{var}(X+Y)=80$, and $\operatorname{var}(X-Y)=40$, find $\operatorname{var}(Y)$ and $\operatorname{cov}(X, Y)$.

Jacob Fry

Numerade Educator

03:18

Problem 44

Suppose that $X$ and $Y$ are continuous random variables with joint density function given by the rule
$$
f(x, y)= \begin{cases}k(x+y) & \text { for } 0 \leq x \leq 2,0 \leq y \leq 2 \\ 0 & \text { elsewhere }\end{cases}
$$
(a) find $k$
(b) find the marginal density functions of $X$ and $Y$
(c) find the conditional density function of $X$, given that $Y=1$
(d) find the conditional density function of $X$, given that $Y=\frac{1}{2}$
(e) ale $X$ and $Y$ independent?
(f) find $E(X), E(Y), E(X Y), \operatorname{cov}(X, Y)$, and $\rho$.

Amany Waheeb

Numerade Educator

Problem 45

For the distribution in Exercise 1, find the first three moments about the origin and the first three moments about the mean.

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Problem 46

Suppose that on a final exam in statistics the mean was 50 and the standard deviation was 10 . Find the following:
(i) the standardized (z) scores of students receiving the following grades: $50,25,0,100,64$.
(ii) the raw grades corresponding to standardized scores of
$$
-2, \quad 2, \quad 1.95, \quad-2.58, \quad 1.65, \quad .33 .
$$

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03:04

Problem 47

For the distribution in Exercise 2, find the distribution of the corresponding standardized random variable, $z$. In what ways are the distributions of $X$ and $z$ similar and in what ways are they dissimilar?

Christopher Stanley

Numerade Educator

00:42

Problem 48

The covariance is not a good measure of the strength of a relationship between two random variables because it depends so much on the units of measurement of the two variables. However, if $X$ and $Y$ are standardized random variables, this problem should be eliminated. Is it?

Lucas Finney

Numerade Educator

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Problem 49

For the distribution of Exercise 1, calculate $P(|X-\mu| \geq 2)$ and compare this with the upper bound for this probability obtained from Tchebycheff's inequality.

Shu Naito

Numerade Educator

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Problem 50

For the distribution of Exercise 2, calculate $P(|X-\mu| \geq 1.5)$ and compare this with a) the upper bound obtained from Tchebycheff's inequality and with b) the upper bound obtained from the stronger form of Tchebycheff's inequality (3.27.3).

Shu Naito

Numerade Educator

01:01

Problem 51

A rough computational check on the accuracy of a standard deviation is that around six times the standard deviation should, in general, include almost the entire range of values for the distribution on which it is based. Do you sce any reason why this rule should work? Must it be true?

Trinity Steen

Numerade Educator