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A First Course in Probability

Sheldon M. Ross

$Math Studyset 魅影直播 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 7: Q.7.64 (page 357)

Type $i$ light bulbs function for a random amount of time having mean $渭 i$ and standard deviation $蟽 i, i = 1, 2$ . A light bulb randomly chosen from a bin of bulbs is a type $1$ bulb with probability $p$ and a type $2$ bulb with probability $1 鈭� p$ . Let $X$ denote the lifetime of this bulb. Find
(a) $E [X]$ ;
(b) $V a r (X)$ .

Short Answer

Expert verified

The values of $E [X]$ and $V a r (X)$ is

a) $E [X] = p 渭_{1} + (1 鈭� p) 渭_{2}$

b) $Var (X) = [p 蟽_{1}^{2} + (1 鈭� p) 蟽_{2}^{2}] + p (1 鈭� p) {(渭_{1} 鈭� 渭_{2})}^{2}$

Step by step solution

01

Step 1:Given Information(part a)

Given that $X$ denotes the lifetime of the bulb.

$Y = \{\begin{cases} 1 鈥呪赌呪赌呪赌� type 1 bulb with probability p \\ 2 鈥呪赌呪赌呪赌� type 2 bulb with probability (1 鈭� p) \end{cases}$

role="math" localid="1647349813765" $P (Y = 1) = p$ , $P (Y = 2) = (1 鈭� p)$

02

Step 2:Explanation(part a)

$E [X] = E (X 鈭� Y = 1) 鈰� P (Y = 1) + E (X 鈭� Y = 2) 鈰� P (Y = 2)$

$= p 渭_{1} + (1 鈭� p) 渭_{2}$

03

Step 3:Final Answer(part a)

The $E [X]$ is

\begin{aligned} E [X] = p 渭_{1} + (1 鈭� p) 渭_{2} \end{aligned}

04

Step 4:Given Information(part b)

Given that $X$ as the lifetime of this bulb.

$Var (X) = E [X^{2}] 鈭� (E [X])^{2}$

05

Step 5:Explanation(Step 5)

$E [X^{2}] = E (X^{2} 鈭� Y = 1) 鈰� P (Y = 1) + E (X^{2} 鈭� Y = 2) 鈰� P (Y = 2)$

$= p (渭_{1}^{2} + 蟽_{1}^{2}) + (1 鈭� p) (渭_{2}^{2} + 蟽_{2}^{2})$

$= [p 蟽_{1}^{2} + (1 鈭� p) 蟽_{2}^{2}] + [p 渭_{1}^{2} + (1 鈭� p) 渭_{2}^{2}]$

$Var (X) = [p 蟽_{1}^{2} + (1 鈭� p) 蟽_{2}^{2}] + [p 渭_{1}^{2} + (1 鈭� p) 渭_{2}^{2}] 鈭� p 渭_{1} + {[(1 鈭� p) 渭_{2}]}^{2}$

$= [p 蟽_{1}^{2} + (1 鈭� p) 蟽_{2}^{2}] + p (1 鈭� p) [渭_{1}^{2} + 渭_{2}^{2} + 2 渭_{1} 渭_{2}]$

$= [p 蟽_{1}^{2} + (1 鈭� p) 蟽_{2}^{2}] + p (1 鈭� p) {(渭_{1} 鈭� 渭_{2})}^{2}$

06

Step 6:Final Answer(part b)

The $V a r (X)$ is

Var (X) = [p 蟽_{1}^{2} + (1 鈭� p) 蟽_{2}^{2}] + p (1 鈭� p) {(渭_{1} 鈭� 渭_{2})}^{2}

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Most popular questions from this chapter

There are two misshapen coins in a box; their probabilities for landing on heads when they are flipped are, respectively, . $4$ and . $7$ . One of the coins is to be randomly chosen and flipped 10 times. Given that two of the first three flips landed on heads, what is the conditional expected number of heads in the $10$ flips?

Use Table $7.2$ to determine the distribution of ${鈭�}_{i = 1}^{n} X_{i}$ when $X_{1}, 鈥�, X_{n}$ are independent and identically distributed exponential random variables, each having mean $1 / 位$ .

If $Y = a X + b$ where a and b are constants, express the moment generating function of $Y$ in terms of the moment generating function of $X$ .

A pond contains $100$ fish, of which $30$ are carp. If $20$ fish are caught, what are the mean and variance of the number of carp among the $20$ ?What assumptions are you making?

A die is rolled twice. Let X equal the sum of the outcomes, and let Y equal the first outcome minus the second.

Compute $C o v (X, Y) .$

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