For any event A we can define tile indicator variable Xa as

For any event A, we can define tile indicator variable Xa as follows Show that two events A and B are independent exactly when their indicator variables Xa and Xb are uncorrelated (have correlation coefficient p = 0).

Solution

They are independent if and only if Pr(X_A=1 &X_B=1)=pq , where X_A=1 with probability p

X_A=0  with probability 1- p

where XB=1 with probability q

  XB=0  with probability 1- q

covariance between X_A and X_B COV(X_AX_B)=E(X_AX_B) - E(X_A)E(X_B) =E(X_AX_B) -pq

E(X_AX_B) =0.Pr(X_AX_B=0) +1.Pr(X_AX_B=1)=

X_AX_B=1 , if and only if X_B=1 and X_B=1 Pr(X_AX_B=1)

So if the covariance is 0 or correlation is 0 then Pr(X_A =1 and X_B=1)=pq or Pr(X_AX_B=1)=pq

therefore two events A and B are independent when their indicator variables X_A , X_B are uncorrelated.