Got Stuck with these probability problems

Question

I tried my best to solve 'em , but after waiting a few sheets of paper , I got nothing on me . A litle help from you guys might do the trick , Thanks !

https://docs.google.com/file/d/0B0wuQOQQfZf2RUw5eFhkSWlha1k/edit?usp=drive_web

Answer 1

For problem 2: Let's say number of transmits is X, Number of $1$s is Z. Z follows a binomial distribution of prob p and with X tries (X and Z are independent). Make it conditional to X: $P(Z=k)=\sum_{n=0}^{\infty}P(X=n) \binom{n}{k}p^k(1-p)^{n-k}$ Just plug in the distribution of X. On your way, you will encounter the series-form of the exponential function. You will get $P(Z=k)=\frac{e^{-p\lambda}(p\lambda)^k}{k!}$.

For problem 3 remember that P(A, B)=P(A|B)P(B) and take conditional propabilities like $P(T=k)=\sum_j P(T=k, N=j)$