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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⁡(X,Y)=E(XY)−E(X)E(Y)

Suppose X and Y are independent

Cov⁡(X,Y)=E(XY)−E(X)E(Y)

={0×0×P(X=0,Y=0)}+{1×1×P(X=1,Y=1)}

−[{0×P(X=0)}+{1×P(X=1)}][{0×P(Y=0)}+{1×P(Y=1)}]

=P(X=1,Y=1)−P(X=0)P(Y=1)

=P(X=0)P(Y=1)−P(X=0)P(Y=1)

=0

Therefore X and Y are independent

03

Solution 

Now let as consider,

Cov⁡(X,Y)=0

⇒E(XY)=E(X)E(Y)

⇒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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