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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.

Alex M.
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Jayson Humarang
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  • you want to compute the cumulative distribution function of the binomial distribution?! – Stephan Kulla Nov 14 '15 at 20:48
  • yes. It is what the problem is asking for. – Jayson Humarang Nov 14 '15 at 22:10
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    For a continuous random variable $X$ with CDF $F$, $F(X) \sim Uniform([0,1])$. This is unfortunately not true for discrete random variables. What was the exact statement of the problem you were given? – Batman Nov 14 '15 at 22:29
  • @JaysonHumarang No, the question does not ask "to compute the cumulative distribution function of the binomial distribution", please read carefully. Re personal input (there is none in your question at the moment), did you try to solve completely the case n=2, say? – Did Nov 14 '15 at 22:52

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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

Community
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Stephan Kulla
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  • This does not answer the question (I know, the OP said their question is to compute F but it is not, and please do not let us get started on users posting questions they do not understand...). – Did Nov 14 '15 at 22:49