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Let $N$ have a $Bin(100, 1/4)$ distribution. Given $N=n$, flip a fair coin $n$ times, and let $X$ be the number of heads observed. What is the distribution of $X$ given $N=n$. Be sure to provide a range and a proper conditional probability mass function.

Can someone help with the intuition behind joint probability mass function. I feel it to has to be binomial, since the conditional will have two binomials over each other. Is it as simple as $$X|N=n\sim Bin(n,1/2)? $$

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    $\begingroup$ Yes, but write $X \mid N=n \sim Bin(n, 1/2)$, not simply $X\sim\ldots$. $\endgroup$ – Jimmy R. Dec 20 '15 at 18:26
  • $\begingroup$ Edited, thanks. $\endgroup$ – Jim Gross Dec 20 '15 at 18:30
  • $\begingroup$ Yes, your wording is not exactly very nice, eg. what do you mean by "two binomials over each other?" but your solution is correct. No need to calculate the joint probability mass function. The conditional of $X$ given that $N=n$ is pretty much clear. And as said it is exactly the one you have. $\endgroup$ – Jimmy R. Dec 21 '15 at 12:52

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