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If $X_1,X_2,…,X_n∼Bernoulli(p)$

Variance of the average of $X$ is

$Var[S_x/n]=\frac{p(1−p)}{n}$

But if we have sample, where all $X$ are equal, $\hat{p}=1$ (or zero), and estimation of var of mean became equal to 0. Is it correct?

Bogdan Shevchenko
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  • What do you want to do with the estimate? Bayes methods is one option, or use the *rule of three* to get a confidence interval, see https://stats.stackexchange.com/questions/274855/using-rule-of-three-to-obtain-confidence-interval-for-a-binomial-population. If that link doesn't serve you, explain why in a comment. – kjetil b halvorsen Jun 19 '19 at 09:44
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    I think, rule of three is exactly what i want, thank you! – Bogdan Shevchenko Jun 19 '19 at 09:54
  • [Here is a paper](https://www.jstor.org/stable/pdf/27919727.pdf?refreqid=excelsior%3A99799e01f5e85fc28adfa4ab395c7449) with an extended discussion. – kjetil b halvorsen Jun 19 '19 at 10:01

1 Answers1

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When $p=1$ or $p=0$, variance of each individual $X_i$, mean of them, sum of them, or any deterministic function of them will be $0$ because now $X_i$'s are certain (they become constant), i.e. you know what's going to happen. For example, if $p=1$, then all $X$ will be $1$, and the average will still be $1$. One can tell you the result without uncertainty as I did.

gunes
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    I think, i need to correct my question: not $p=0$, but $\hat{p}=0$. So, real p probably is not 1 or 0, and real $Var > 0$, but how we can estimate it? – Bogdan Shevchenko Jun 19 '19 at 09:39
  • What do you want to do with the estimate? Bayes methods is one option, or use the *rule of three* to get a confidence interval, see https://stats.stackexchange.com/questions/274855/using-rule-of-three-to-obtain-confidence-interval-for-a-binomial-population. If that link doesn't serve you, explain why in a comment. – kjetil b halvorsen Jun 19 '19 at 09:43