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Chapter 7: Q.7.36 (page 355)

Let Xbe the number of 1鈥�sand Ythe number of 2'sthat occur in nrolls of a fair die. Compute Cov(X,Y).

Short Answer

Expert verified

The value ofCov(X,Y)is-n36.

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 nrolls of a fair Die

Y=Number of 2's in nrolls of A fair Die

鈭�E(X)=n6

E(Y)=n6

E(XY)=n6脳(n-1)6

=n(n-1)36

鈭�Cov(X,Y)=E(XY)-E(X)路E(Y)

03

Explanation

ComputeCov(X,Y)

=n6路n-16-n6路n6

=n236-n36-n236

=-n36

04

Final Answer 

Hence, the value ofCov(X,Y)is-n36.

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Most popular questions from this chapter

Let Z be a standard normal random variable,and, for a 铿亁ed x, set

X={ZifZ>x0otherwise

Show thatE[X]=12蟺e鈭�x2/2.

Let X1,...be independent with common mean 渭and common variance 蟽2, and set Yn=Xn+Xn+1+Xn+2. For j鈮�0, find CovYn,Yn+j.

Let 蠒be the standard normal distribution function, and let X be a normal random variable with mean 渭 and variance 1. We want to find E[ 蠒(X)]. To do so, let Z be a standard normal random variable that is independent of X, and let

I=1,鈥呪赌呪赌呪赌�ifZ<X0,鈥呪赌呪赌呪赌�ifZ鈮�X

(a) Show that E[I鈭�X=x]=桅(x).

(b) Show that E[桅(X)]=P{Z<X}.

(c) Show that E[桅(X)]=桅渭2.

Hint: What is the distribution of X-Z?

The preceding comes up in statistics. Suppose you are about to observe the value of a random variable X that is normally distributed with an unknown mean 渭 and variance 1, and suppose that you want to test the hypothesis that the mean 渭 is greater than or equal to 0. Clearly you would want to reject this hypothesis if X is sufficiently small. If it results that X = x, then the p-value of the hypothesis that the mean is greater than or equal to 0 is defined to be the probability that X would be as small as x if 渭 were equal to 0 (its smallest possible value if the hypothesis were true). (A small p-value is taken as an indication that the hypothesis is probably false.) Because X has a standard normal distribution when 渭 = 0, the p-value that results when X = x is 蠒 (x). Therefore, the preceding shows that the expected p-value that results when the true mean is 渭 is 蠁渭2 .

For a group of 100 people, compute

(a) the expected number of days of the year that are birthdays of exactly 3 people;

(b) the expected number of distinct birthdays.

7.2. Suppose that Xis a continuous random variable with

density function f. Show that E[IX-a鈭�]is minimized

when ais equal to the median of F.

Hint: Write

E[IX-al]=|x-a|f(x)dx

Now break up the integral into the regions where x<a

and where x>a, and differentiate.
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