Problem 1

X and Y are iid

   $  P(X=i) = P(Y=i) = \frac {1}{2^i}\ ,i = 1,2,3,...  $

Part A

   Find $  P(min(X,Y)=k)\  $

   Let $  Z = min(X,Y)\  $

Then finding the pmf of Z uses the fact that X and Y are iid

   $  P(Z=k) = P(X \ge k,Y \ge k) = P(X \ge k)P(Y \ge k) = P(X \ge k)^2  $

   $  P(Z=k) = \left ( \sum_{i=k}^N \frac {1}{2^i} \right )^2 = \left ( \frac {1}{2^k} \right )^2 = \frac {1}{4^k}  $