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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.7.54 (page 363)

If Z is a standard normal random variable, what is Cov(Z, Z²)?

Short Answer

Expert verified

The covariance of $Z$ and $Z^{2}$ is $C o v (Z, Z^{2}) = 0$ .

Step by step solution

01

Given information

Z is a standard normal random variable

02

Solution

Let $Z^{2}$ be the square of the standard normal variable Z.

The covariance of Z and $Z^{2}$ is calculated below,

$Cov (Z, Z^{2}) = E (Z \cdot Z^{2}) - E (Z) E (Z^{2})$

$= E (Z^{3}) - 0 \times E (Z^{2})$

$= E (Z^{3}) (\begin{array}{l} From the raw moments of \\ Standard normal distribution μ_{3} = 0 \end{array})$

$= 0$

03

Final answer

Therefore the covariance of $Z$ and $Z^{2}$ is $Cov (Z, Z^{2}) = 0$ .

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Most popular questions from this chapter

Consider $n$ independent trials, the $i t h$ of which results in a success with probability $P_{l}$ .

(a) Compute the expected number of successes in the $n$ trials-call it $μ$

(b) For a fixed value of $μ$ , what choice of $P_{1}, \dots, P_{n}$ maximizes the variance of the number of successes?

(c) What choice minimizes the variance?

Repeat Problem 7.68 when the proportion of the population having a value of $λ$ less than $x$ is equal to $1 - e^{- x}$ .

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. We are required to find $P (N = 0) .$

b. We are required to find $P (N = 3)$ .

c. Define $M$ as the number of accidents in a preceding year. As likely as $N$ we are require to find.

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

$f (x, y) = \frac{1}{\sqrt{2 π}} e^{- y} e^{- (x - y)^{2} / 2} 0 < y < \infty$ ,

$- \infty < x < \infty$

(a) Compute the joint moment generating function of $X$ and $Y$ .

(b) Compute the individual moment generating functions.

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?

Show that $E [{(X - a)}^{2}]$ is minimized at $a = E [X]$ .

See all solutions

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