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Chapter 7: Q.7.29 (page 365)

Suppose that X and Y are both Bernoulli random variables. Show that X and Y are independent if and only if Cov(X, Y) = 0.

Short Answer

Expert verified

It is clear from the calculation that the X and Y are independent Variables.

Step by step solution

01

Given information

X and Y are both Bernoulli random variables.

02

Solution

First we need to calculate the Cov鈦�(X,Y)=E(XY)鈭�E(X)E(Y)

Suppose X and Y are independent

Cov鈦�(X,Y)=E(XY)鈭�E(X)E(Y)

={0脳0脳P(X=0,Y=0)}+{1脳1脳P(X=1,Y=1)}

鈭�[{0脳P(X=0)}+{1脳P(X=1)}][{0脳P(Y=0)}+{1脳P(Y=1)}]

=P(X=1,Y=1)鈭�P(X=0)P(Y=1)

=P(X=0)P(Y=1)鈭�P(X=0)P(Y=1)

=0

Therefore X and Y are independent

03

Solution 

Now let as consider,

Cov鈦�(X,Y)=0

鈬�E(XY)=E(X)E(Y)

鈬�P(X=1,Y=1)=P(X=0)P(Y=1)

So, X and Y are independent.

04

Final answer

It is clear that the X and Y are independent Variables.

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Most popular questions from this chapter

Let X1,X2,鈥�,Xnbe independent random variables having an unknown continuous distribution function Fand let Y1,Y2,鈥�,Ymbe independent random variables having an unknown continuous distribution function G. Now order those n+mvariables, and let

Ii=1鈥呪赌呪赌呪赌�if theith smallest of then+m鈥呪赌呪赌呪赌�variables is from theXsample0鈥呪赌呪赌呪赌�otherwise

The random variable R=鈭�i=1n+miIiis the sum of the ranks of the Xsample and is the basis of a standard statistical procedure (called the Wilcoxon sum-of-ranks test) for testing whether Fand Gare identical distributions. This test accepts the hypothesis that F=Gwhen Ris neither too large nor too small. Assuming that the hypothesis of equality is in fact correct, compute the mean and variance of R.

Hint: Use the results of Example 3e.

If Xand Y are independent and identically distributed with mean 渭and variance 蟽2, find

E(X鈭�Y)2

The best linear predictor of Ywith respect toX1and X2is equal to a+bX1+cX2, where a, b, and care chosen to minimizeEY-a+bX1+cX22 Determine a, b, and c.

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(a) 2 aces;

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