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A First Course in Probability

Sheldon M. Ross

$Math Studyset Vaia Explanations$ Math

9th Edition · 432 pages

ISBN: 9780321794772

Chapter 7: Q.7.3 (page 354)

If $X$ and Y are independent and identically distributed with mean $μ$ and variance $σ 2$ , find
$E [(X - Y)^{2}]$

Short Answer

Expert verified

The value of $E [(X - Y)^{2}]$ is

$E [(X - Y)^{2}] = 2 σ^{2}$

Step by step solution

01

Step 1:Given Information

$X$ and $Y$ are autonomous and identically allocated with mean $μ$ and variance $σ^{2}$ .

02

Step 2:Explanation

$E [X] = E [Y] = μ$

$Var (X) = Var (Y) = σ^{2}$

$Var (X) = E [X^{2}] - (E [X])^{2}$

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

$= σ^{2} + μ^{2}$

$X$ and $Y$ are independent,

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

$= E [X^{2}] - 2 E [X] E [Y] + E [Y^{2}]$

$= (σ^{2} + μ^{2}) - 2 μ μ + (σ^{2} + μ^{2})$

$= 2 σ^{2} + 2 μ^{2} - 2 μ^{2}$

$= 2 σ^{2}$

03

Step 3:Final Answer

$E [(X - Y)^{2}] = 2 σ^{2}$

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Most popular questions from this chapter

Consider a gambler who, at each gamble, either wins or loses her bet with respective probabilities $p$ and $1 - p$ . A popular gambling system known as the Kelley strategy is to always bet the fraction $2 p - 1$ of your current fortune when $p > 12$ . Compute the expected fortune after $n$ gambles of a gambler who starts with $x$ units and employs the Kelley strategy.

Let $X_{1}, X_{2}, \dots$ be a sequence of independent random variables having the probability mass function $P \{X_{n} = 0\} = P \{X_{n} = 2\} = 1 / 2, n \geq 1$

The random variable $X = \sum_{n = 1}^{\infty} X_{n} / 3^{n}$ is said to have the Cantor distribution.

Find $E [X]$ and $V a r (X) .$

Cards from an ordinary deck are turned face up one at a time. Compute the expected number of cards that need to be turned face up in order to obtain

(a) 2 aces;

(b) 5 spades;

(c) all 13 hearts.

In Example $2$ h,say that i and $j, i \neq j$ , form a matched pair if i chooses the hat belonging to j and j chooses the hat belonging to i. Find the expected number of matched pairs.

Show that $X$ is stochastically larger than $Y$ if and only if $E [f (X)] \geq E [f (Y)]$

for all increasing functions $f .$ .

Hint: Show that $X \geq_{st} Y$ , then $E [f (X)] \geq E [f (Y)]$ by showing that $f (X) \geq_{s t} f (Y)$ and then using Theoretical Exercise 7.7. To show that if $E [f (X)] \geq E [f (Y)]$ for all increasing functions $f$ , then $P {X > t} \geq P {Y > t}$ , define an appropriate increasing function $f$ .

See all solutions

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