Question

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

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

Step-by-step solution

Given information

X and Y are both Bernoulli random variables.

Solution

First we need to calculate the $\mathrm{Cov}(X,Y)=E\left(XY\right)-E\left(X\right)E\left(Y\right)$

Suppose X and Y are independent

$\mathrm{Cov}(X,Y)=E\left(XY\right)-E\left(X\right)E\left(Y\right)$

$=\{0\times 0\times P(X=0,Y=0\left)\right\}+\{1\times 1\times P(X=1,Y=1\left)\right\}$

$-\left[\right\{0\times P(X=0)\}+\{1\times P(X=1)\left\}\right]\left[\right\{0\times P(Y=0)\}+\{1\times P(Y=1)\left\}\right]$

$=P(X=1,Y=1)-P(X=0)P(Y=1)$

$=P(X=0)P(Y=1)-P(X=0)P(Y=1)$

$=0$

Therefore X and Y are independent

Solution 

Now let as consider,

$\mathrm{Cov}(X,Y)=0$

$\Rightarrow E\left(XY\right)=E\left(X\right)E\left(Y\right)$

$\Rightarrow P(X=1,Y=1)=P(X=0)P(Y=1)$

So, X and Y are independent.

Final answer

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