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}  $

Part B

• Find $ P(X=Y)\ $

   Noting that X and Y are iid and summing across all possible i,
       $  P(X=Y) = \sum_{i=1}^\infty P(X=i, Y=i) = \sum_{i=1}^\infty P(X=i)P(Y=i) = \sum_{i=1}^\infty \frac {1}{4^i} = \frac {1}{3}  $

Part C

• Find $ P(Y>X)\ $

   Again noting that X and Y are iid and summing across all possible i,
       $  P(Y>X) = \sum_{i=1}^\infty P(Y>i, X=i) = \sum_{i=1}^\infty P(Y>i)P(x=i) $

   Next, find $  P(Y<i)\  $
       $  P(Y>i) = 1 - P(Y \le i)  $

       $  P(Y \le i) = \sum_{i=1}^\infty \frac {1}{2^i} = 1 + \frac {1}{2^i}  $

     $  \therefore P(Y>i) = \frac {1}{2^i}  $

   Plugging this result back into the original expression yields
       $  P(Y<X) = \sum_{i=1}^\infty \frac {1}{4^i} = \frac {1}{3}  $

Part D

• Find $ P(Y=kX)\ $

   Noting that X and Y are iid and summing over all possible combinations one arrives at
       $  P(Y=kX) = \sum_{i=1}^\infty i = 1^\infty P(Y=ki, X=i) = \sum_{i=1}^\infty P(Y=ki)P(X=i)  $

   Thus,
       $  P(Y=kX) = \sum_{i=1}^\infty \frac {1}{2^{ki}} \frac {1}{2^i} = \sum_{i=1}^\infty \frac {1}{2^{(k+1)i}} = \frac {1}{2^{(k+1)}-1}  $