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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 8: Q. 8.21 (page 392)

Let $X$ be a non-negative random variable. Prove that
$E [X] 鈮� {(E [X^{2}])}^{1 / 2} 鈮� {(E [X^{3}])}^{1 / 3} 鈮� 鈰�$

Short Answer

Expert verified

Apply Lyapunov's inequality (proof is given inside) to a random variable $0 < 1 < 2 < 3 < 鈰�$ .

Step by step solution

01

Given Information.

Let $X$ be a nonnegative random variable.

02

Explanation.

Suppose that $X$ is a nonnegative random variable. The relationship

$E [X] 鈮� {(E [X^{2}])}^{1 / 2} 鈮� {(E [X^{3}])}^{1 / 3} 鈮� 鈰�$

we need to prove the consequence of Lyapunov's inequality, so let's prove this general result.

Lyapunov's inequality: If $0 < s < t, then {(E [| X |^{s}])}^{1 / s} 鈮� {(E [| X |^{t}])}^{1 / t} .$

Proof: Consider the function $f (y) = | y |^{r}, r > 1$ . The function $f (y)$ is a convex function. Then, to Jensen's inequality,

$E [f (Y)] 鈮� f (E [Y]) 鈬� E [| Y |^{r}] 鈮� | E [Y] |^{r} r = \frac{t}{s} > 1, let Y = | X |^{s} E [| X |^{t}] 鈮� {(E [| X |^{s}])}^{t / s}$

| taking the $t$ root of each side we get:

${(E [| X |^{t}])}^{1 / t} 鈮� {(E [| X |^{s}])}^{1 / s} .$

Since $X 鈮� 0$ we have that $| X | = X$ . Apply Lyapunov's inequality to a random variable $X$ =localid="1649893415983" $0 < 1 < 2 < 3 < 鈰�$ .

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Most popular questions from this chapter

Each of the batteries in a collection of $40$ batteries is equally likely to be either a type A or a type B battery. Type A batteries last for an amount of time that has a mean of $50$ and a standard deviation of $15$ ; type B batteries last for a mean of $30$ and a standard deviation of 6.

(a) Approximate the probability that the total life of all $40$ batteries exceeds $1700$

(b) Suppose it is known that $20$ of the batteries are type A and $20$ are type B. Now approximate the probability that the total life of all $40$ batteries exceeds $1700 .$

It $X$ is a Poisson random variable with a mean $位$ , showing that for $i < 位$ ,

$P {X 鈮� i} 鈮� \frac{e^{- 位} (e 位)^{i}}{i^{i}}$

Suppose that the number of units produced daily at factory A is a random variable with mean $20$ and standard deviation $3$ and the number produced at factory B is a random variable with mean $18$ and standard deviation of $6$ . Assuming independence, derive an upper bound for the probability that more units are produced today at factory B than at factory A.

If, $X$ is a nonnegative random variable with a mean of $25$ , what can be said about

role="math" localid="1649836663727" $(a) E [X^{3}] ?$

$(b) E [鈭� X] ?$

$(c) E [\log X] ?$

role="math" localid="1649836643047" $(d) E [e^{- X}] ?$

From past experience, a professor knows that the test score taking her final examination is a random variable with a mean of $75$ .

$(a)$ Give an upper bound for the probability that a student鈥檚 test score will exceed $85$ .

$(b)$ Suppose, in addition, that the professor knows that the variance of a student鈥檚 test score is equal $25$ . What can be said about the probability that a student will score between $65$ and $85$ ?

$(c)$ How many students would have to take the examination to ensure a probability of at least $. 9$ that the class average would be within $5$ of $75$ ? Do not use the central limit theorem.

See all solutions

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