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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

A fair die is successively rolled. Let X and Y denote, respectively, the number of rolls necessary to obtain a 6 and a 5. Find

(a) E[X];

(b) E[X鈭�Y=1];

(c) E[X鈭�Y=5];

Consider the following dice game: A pair of dice is rolled. If the sum is7,then the game ends and you win 0.If the sum is not 7,then you have the option of either stopping the game and receiving an amount equal to that sum or starting over again. For each value ofi,i=2,...,12, find your expected return if you employ the strategy of stopping the first time that a value at least as large as i appears. What value ofileads to the largest expected return? Hint: Let Xidenote the return when you use the critical value i.To computeE[Xi], condition on the initial sum.

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

If E[X]=1and Var(X)=5find

(a)E[(2+X2)]

(b)Var(4+3X)

Let X1,X2,鈥�,Xn be independent and identically distributed positive random variables. For k鈮�n findE鈭�i=1kXi鈭�i=1nXi.
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