If X1 and X2 are jointly distributed random variables their

If X₁ and X₂ are jointly distributed random variables, their covariance is dened as cov(X₁,X₂) = E[X₁X₂] - E[X₁]E[X₂]. Draw two cards in succession from a standard deck Let X_i = 1 if the i^th draw is a heart and let X_i=0 otherwise. Find he covariance between X₁ and X₂ if the cards are drawn with replacement.

Solution

Probability distribution for X1 and X2:

P(X1=1 and X2=1) = 13/52 * 13/52 = 1/16

P(X1=1 and X2=0) = 13/52 * 39/52 = 3/16

P(X1=0 and X2=1) = 39/52 * 13/52 = 3/16

P(X1=0 and X2=0) = 39/52 * 39/52 = 9/16

E(X1*X2) = 1*1/16 + 0*3/16 + 0*3/16 + 0*9/16 = 1/16

P(X1=0) = 39/52 = 3/4

P(X1=1) = 13/52 = 1/4

E(X1) = 0*3/4 + 1*1/4 = 1/4

P(X2=0) = 39/52 = 3/4

P(X2=1) = 13/52 = 1/4

E(X2) = 0*3/4 + 1*1/4 = 1/4

Cov(X1,X2) = 1/16 - (1/4*1/4) = 0