Let $Y \sim \mathrm{Bin}(n, q)$ and $X \sim \mathrm{Bin}(Y, p)$, is there an expression for the distribution of $X/Y$?
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3Binomial distributions take the value $0$ with positive probability, so you may be looking at $\frac00$ when $Y=0$, which has probability $(1-q)^n$. You may want to define this case to be $0$ or $1$ or $p$ or something else but you do need to handle it – Henry Apr 07 '22 at 08:26
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https://stats.stackexchange.com/questions/153951/ ran into the same difficulty and therefore no answer could be supplied. https://stats.stackexchange.com/questions/458789 solves a related statistical problem indirectly -- perhaps that would be relevant to your application? – whuber Apr 07 '22 at 13:53
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@Henry good point – geometrikal Apr 07 '22 at 20:31
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@whuber thanks for that. The application is related to machine learning. Say I have an image classifier that I've trained. I randomly select an images from a test set of which I know the classes (Y is number of images for a class that occurs with frequency q from n selections). I then classify them with the model and record true / false (X where p is the probability the model classifies correctly) and calculate the accuracy (X/Y). I hoping to get an idea for how many (n) images I need to select to have the accuracy occur within a certain confidence interval. – geometrikal Apr 07 '22 at 20:38