Question

9. Let X and Y be Bernouilli random variables with joint distribution: Pr(X = 1 and Y = 1) = 0.42, Pr(X = 1 and Y = 0) = 0.18, Pr(X = 0 and Y = 1) = 0.28, Pr(X = 0 and Y = 0) = 0.12 Determine whether or not X and Y are independent. Determine the covariance and correlation for X and Y. Can you infer anything from this correlation?

Answer 1

The Joint distribution of ( X , Y ) is :

X\Y	0	1	Total
0	0.12	0.28	0.40
1	0.18	0.42	0.60
Total	0.30	0.70	1

Marginal of X is :

X	0	1
P ( X = x )	0.40	0.60

Marginal of Y is :

Y	0	1
P ( Y = y )	0.30	0.70

X and Y are Independent as :

P(X=x, Y=y)= P(X = ).P(Y = y) for all and y

• 
  P(X 0, Y = 0) = P(X= 0).P(Y 0)=0.4 0.3 0.12
  
• 
  P(X 0, Y 1)= P(X= 0).P(Y 1)0.4 0.7 0.28
  
• 
  P(X 1, Y 0) = P(X = 1).P(Y 0)=0.6 0.3 0.18
  
• 
  P(X 1, Y 1)= P(X = 1).P(Y 1)0.6 0.7 0.42
  

$Cov(X,Y)=E(XY)-E(X)E(Y)$

• $E(X)=\sum x.P(X=x)=0(0.4)+1(0.6)=0.6$
• 
  E(Y) y.P(Y y)0(0.3)1 (0.7) 0.7
  
• 
  E(XY) ayP (X ,Y y)
  

(0) (0) (0.12) (0)(1)(0.28) (1)(0) (0.18 )(1)(1) (0.42)

$=0.42$

$\Rightarrow Cov(X,Y)=0.42-0.6(0.7)=0$

$\Rightarrow Corr(X,Y)=\rho=\frac{Cov(X,Y)}{\sqrt{V(X).V(Y)}}=0$

X and Y are two independent Random Variables . Thus , their correlation must be zero .

The Joint Distribution is equal to the product of Marginals .

Also , Independence implies correlation is zero but correlation of zero doesn't necessarily implies independency .