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A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 7: Q.7.33 (page 354)

If $E [X] = 1$ and $V a r (X) = 5$ find
(a) $E [(2 + X^{2})]$
(b) $V a r (4 + 3 X)$

Short Answer

Expert verified

a) The value of $E [(2 + X^{2})]$ is $14$ .

b) The value of $Var (4 + 3 X)$ is $45 .$

Step by step solution

01

Given information (Part a)

$E (X) = 1$

$V a r (X) = 5$

Then $E [(2 + X^{2})] = ?$

02

Explanation (Part a)

We know that

$E (X^{2}) = Var (X) + (E (X))^{2}$

$= 5 + 1$

$= 6$

Calculate the expected value of $E [(2 + X^{2})]$

$E [(2 + X)^{2}] = E [4 + X^{2} + 4 X]$

$= 4 + 6 + 4$

$= 14$

03

Final Answer (Part a)

The value of $E [(2 + X^{2})]$ is $14 .$

04

Given information (Part b)

$E (X) = 1$

$Var (X) = 5$

Then $Var (4 + 3 x) = ?$

05

Explanation (Part b)

We know that,

$E (X^{2}) = Var (X) + (E (X))^{2}$

$= 5 + 1$

role="math" localid="1647519062218" $= 6$

Calculate the variance.

$V a r (4 + 3 X) = 0 + 9 V a r (X)$

$= (9 脳 5)$

$= 45$

06

Final Answer (Part b)

Hence, the value of $V a r (4 + 3 x)$ is $45 .$

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Most popular questions from this chapter

A fair die is rolled $10$ times. Calculate the expected sum of the $10$ rolls.

A bottle initially contains m large pills and n small pills. Each day, a patient randomly chooses one of the pills. If a small pill is chosen, then that pill is eaten. If a large pill is chosen, then the pill is broken in two; one part is returned to the bottle (and is now considered a small pill) and the other part is then eaten.

(a) Let X denote the number of small pills in the bottle after the last large pill has been chosen and its smaller half returned. Find E[X].

Hint: De铿乶e n + m indicator variables, one for each of the small pills initially present and one for each of the small pills created when a large one is split in two. Now use the argument of Example $2$ m.

(b) Let Y denote the day on which the last large pills chosen. Find E[Y].

Hint: What is the relationship between X and Y?

The joint density function of $X$ and $Y$ is given by

$f (x, y) = \frac{1}{y} e^{- (y + x / y)}, x > 0, y > 0$

Find $E [X], E [Y]$ and show that $C o v (X, Y) = 1$

Let $X_{1}, X_{2}, 鈥�, X_{n}$ be independent random variables having an unknown continuous distribution function $F$ and let $Y_{1}, Y_{2}, 鈥�, Y_{m}$ be independent random variables having an unknown continuous distribution function $G$ . Now order those $n + m$ variables, and let

$I_{i} = \{\begin{cases} 1 鈥呪赌呪赌呪赌� if the i th smallest of the n + m \\ 鈥呪赌呪赌呪赌� variables is from the X sample \\ 0 鈥呪赌呪赌呪赌� otherwise \end{cases}$

The random variable $R = {鈭�}_{i = 1}^{n + m} i I_{i}$ is the sum of the ranks of the $X$ sample and is the basis of a standard statistical procedure (called the Wilcoxon sum-of-ranks test) for testing whether $F$ and $G$ are identical distributions. This test accepts the hypothesis that $F = G$ when $R$ is neither too large nor too small. Assuming that the hypothesis of equality is in fact correct, compute the mean and variance of $R$ .

Hint: Use the results of Example 3e.

Suppose that $X_{i}, i = 1, 2, 3$ , are independent Poisson random variables with respective means $位_{i}, i = 1, 2, 3$ . Let $X = X_{1} + X_{2}$ and $Y = X_{2} + X_{3}$ . The random vector $X, Y$ is said to have a bivariate Poisson distribution.
( $a$ ) Find $E [X]$ and $E [Y]$ .
( $b$ ) Find $Cov (X, Y)$ .
( $c$ ) Find the joint probability mass function $P {X = i$ , $Y = j}$ .

See all solutions

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