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First time posting here.

Suppose I have a random variable $X$ and after various algebraic rewriting and calculus I have arrived at a closed form of $\phi = \mathbb{E}[X]$. However, I am not totally confident in the algebraic manipulation so I want to verify the result by sampling.

I can sample directly from $X$ and I also have a closed form of $\mathbb{Var}[X]$ but $X$ is definitely not normally distributed. Is there any statistical test without the assumption of normality that I can perform to get a probability that $\mathbb{E}[X] = \phi$ using only samples drawn i.i.d. from $X$?

Obviously I can just sample $X$ and compare the mean to $\phi$ but this doesn't give me a scientific metric of 'likelihood of correctness' other than simply saying 'look, the error is small!'

Atkrye
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With a huge sample, one can assume normality not of the variable itself, but of the distribution of its sample mean $\bar{X}$ (the estimator of the variable's mean $\mu$).

If you are unsure of the variable's variance $\sigma^2$ as much as of the mean, you may just run Student's t-test and get a confidence interval for the mean $\mu$.

Alex
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  • So the methodology would be to sample N sets of M samples, to produce N estimations of $\mu$, and then run a t-test under the assumption that the random variable of the estimator is normally distributed?

    Why did you write Pearson's t-test? Is that a mistake? or something I'm not familiar with?

     – Atkrye Oct 06 '22 at 00:30
  • Thanks, by the way! – Atkrye Oct 06 '22 at 00:30
  • Nope, you take one largest sample your laptop's memory can accommodate and feed it to Pearson's t-test – Alex Oct 06 '22 at 00:32
  • Wrong naming, corrected ) – Alex Oct 06 '22 at 00:49
  • ahh okay, thanks! – Atkrye Oct 06 '22 at 01:09
  • This will be valid assuming $\sigma^2$ is finite and not terrifically large. It can fail without warning otherwise. It therefore is always a good idea to look at the shape of the distribution of $X$ to ensure you're not in such a bad situation. BTW, a t test is pointless with a "huge" sample. Just compute the Z-score. It's also a very good idea to repeat the entire sampling procedure a few times to see whether any unusually large Z-scores emerge (greater than $3,$ more or less), because that would indicate a problem. See the code at https://stats.stackexchange.com/a/218700/919, e.g. – whuber Oct 06 '22 at 15:05