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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. 7.4 (page 359)

Let $X$ be a random variable having finite expectation $μ$ and variance $σ^{2}$ , and let $g (*)$ be a twice differentiable function. Show that
$E [g (X)] \approx g (μ) + \frac{g^{''} (μ)}{2} σ^{2}$
Hint: Expand $g (\cdot)$ in a Taylor series about $μ$ . Use the first
three terms and ignore the remainder.

Short Answer

Expert verified

The random variable is showed as $E [g (X)] \approx g (μ) + \frac{g^{''} (μ)}{2} σ^{2}$ having finite expectation.

Step by step solution

01

Given Information

The finite expectation of variance and twice differentiable function $g (\cdot)$ .

02

Explanation

If $n ⩾ 0$ is an integer and $g$ is a function which is $n$ times continuously differentiable on the closed interval $[a, x]$ and $n + 1$ times differentiable on the open interval $(a, x)$ , then we have:

$g (x) = g (a) + \frac{g^{'} (a)}{1!} (x - a) + \frac{g^{''} (a)}{2!} (x - a)^{2} + \dots + \frac{g^{(n)} (a)}{n!} (x - a)^{n} + R_{n}$

The remainder term $R n$ depends on $x$ and is small if $x$ is close enough to $a$ .

03

Explanation

Expand $g (\cdot)$ in Taylor polynomial:

$g (x) = g (μ) + \frac{g^{'} (μ)}{1!} (x - μ) + \frac{g^{''} (μ)}{2!} (x - μ)^{2} + R_{n}$

Expected value of both side is,

$[E g (x)] = g (μ) + 0 + \frac{g^{''} (μ)}{2!} σ^{2} + R_{n} (σ^{2}), E (x - μ) = 0$

$[E g (x)] \approx g (μ) + \frac{g^{''} (μ)}{2!} σ^{2}$ .

04

Final answer

The random variable is showed as $[E g (x)] \approx g (μ) + \frac{g^{''} (μ)}{2!} σ^{2}$ having finite expectation.

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Most popular questions from this chapter

Let $X$ be the number of $1 ’ s$ and $Y$ the number of $2' s$ that occur in $n$ rolls of a fair die. Compute $C o v (X, Y)$ .

Let $X_{1}, X_{2}, \dots$ be a sequence of independent and identically distributed continuous random variables. Let $N \geq 2$ be such that

$X_{1} \geq X_{2} \geq \dots \geq X_{N - 1} < X_{N}$

That is, $N$ is the point at which the sequence stops decreasing. Show that $E [N] = e$ .

Hint: First find $P {N \geq n}$ .

Urn $1$ contains $5$ white and $6$ black balls, while urn $2$ contains $8$ white and $10$ black balls. Two balls are randomly selected from urn $1$ and are put into urn $2$ . If $3$ balls are then randomly selected from urn $2$ , compute the expected number of white balls in the trio.

Hint: LetXi = $1$ if the i th white ball initially in urn $1$ is one of the three selected, and let Xi = $0$ otherwise. Similarly, let Yi = $1$ if the i the white ball from urn $2$ is one of the three selected, and let Yi = $0$ otherwise. The number of white balls in the trio can now be written as $\sum_{1}^{5} X_{i} + \sum_{1}^{8} Y_{i}$

The number of accidents that a person has in a given year is a Poisson random variable with mean $λ$ ̣ However, suppose that the value of $λ$ changes from person to person, being equal to $2$ for $60$ percent of the population and $3$ for the other $40$ percent. If a person is chosen at random, what is the probability that he will have

(a) $0$ accidents and,

(b) Exactly $3$ accidents in a certain year? What is the conditional probability that he will have $3$ accidents in a given year, given that he had no accidents the preceding year?

$A$ There are $n + 1$ participants in a game. Each person independently is a winner with probability $p$ . The winners share a total prize of 1 unit. (For instance, if $4$ people win, then each of them receives $1 4$ , whereas if there are no winners, then none of the participants receives anything.) Let A denote a specified one of the players, and let $X$ denote the amount that is received by $A$ .

(a) Compute the expected total prize shared by the players.

(b) Argue that role="math" localid="1647359898823" $E [X] = \frac{1 - {(1 - p)}^{n + 1}}{n + 1}$ .

(c) Compute E[X] by conditioning on whether is a winner, and conclude that role="math" localid="1647360044853" $E [(1 + B)^{- 1}] = \frac{1 - {(1 - p)}^{n + 1}}{(n + 1) p}$ when $B$ is a binomial random variable with parameters $n$ and $p$

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