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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.7.36 (page 355)

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 $C o v (X, Y)$ .

Short Answer

Expert verified

The value of $Cov (X, Y)$ is $\frac{- n}{36} .$

Step by step solution

01

Given Information

Number of $1' s = X$

Number of $2' s = Y$

Fair die rolls $= n$

02

Explanation

Calculate the covariance of $(X, Y)$

$X =$ Number of 1's in $n$ rolls of a fair Die

$Y =$ Number of 2's in $n$ rolls of A fair Die

$∴ E (X) = \frac{n}{6}$

$E (Y) = \frac{n}{6}$

$E (XY) = \frac{n}{6} \times \frac{(n - 1)}{6}$

$= \frac{n (n - 1)}{36}$

$∴ Cov (X, Y) = E (X Y) - E (X) \cdot E (Y)$

03

Explanation

Compute $Cov (X, Y)$

$= \frac{n}{6} \cdot \frac{n - 1}{6} - \frac{n}{6} \cdot \frac{n}{6}$

$= (\frac{n^{2}}{36} - \frac{n}{36}) - \frac{n^{2}}{36}$

$= \frac{- n}{36}$

04

Final Answer

Hence, the value of $Cov (X, Y)$ is $\frac{- n}{36} .$

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Most popular questions from this chapter

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.

Let $X$ be the value of the first die and $Y$ the sum of the values when two dice are rolled. Compute the joint moment generating function of $X$ and $Y$ .

Consider $n$ independent flips of a coin having probability $p$ of landing on heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance, if $n = 5$ and the outcome is $H H T H T$ , then there are $3$ changeovers. Find the expected number of changeovers. Hint: Express the number of changeovers as the sum of $n - 1$ Bernoulli random variables.

Suppose that the expected number of accidents per week at an industrial plant is $5$ . Suppose also that the numbers of workers injured in each accident are independent random variables with a common mean of $2.5$ . If the number of workers injured in each accident is independent of the number of accidents that occur, compute the expected number of workers injured in a week .

A set of $1000$ cards numbered 1 through $1000$ is randomly distributed among $1000$ people with each receiving one card. Compute the expected number of cards that are given to people whose age matches the number on the card.

See all solutions

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