Let $X$ be a random variable with probability mass function $p_k= \binom n k p^k (1-p)^{n-k}$ (binomial). If $F$ is the corresponding distribution function, find the distribution of $F(X)$.
I know for certain that I need to get the sum of $k$ binomial random variables, however I'm having a hard time in doing it. I read from books that it will be needing of mathematical induction. Can somebody help me? Thanks in advance.
So far as I know there is no possibility to give a simple form for $F(x) = \sum_{k \le x} \binom nk p^k (1-p)^{n-k}$ or to summarize the sum. But you can use the theorem by de Moivre-Laplace which is a special case of the central limit theorem. With the Edeworth expansion even better approximations are possible. See the answer of Where is my error in finding the edgeworth expansion of the binomial distribution? for the first term of the Edgeworth expansion.
Update: Maybe you are also looking for https://en.wikipedia.org/wiki/Binomial_distribution#Sums_of_binomials