Question:
The probability that an error is introduced into a packet is $\alpha$. Messages, consisting of one or more packets, are received at a node. Given that a message has been received free of errors, what is the expected number of packets in it? Assume that packets are never lost and that messages of all lengths are equally likely.
I know I need to find the expected value and use the uniform distribution, but since it could range from length = 1 to infinity I'm unsure. What I'm starting with: $$E[X]=\sum_{k=1}^\infty\alpha^kf_X(k)$$
Since its a uniform distribution I thought $f_X(k)$ would be a constant so I have: $$E[X]=f_X(k)\big(\sum_{k=1}^\infty\alpha^k\big)$$ $$E[X]=f_X(k)*\frac{\alpha}{1-\alpha}$$
Any hints on what to do next (how to compute the pdf when infinity is involved) or corrections would be appreciated.