Statistical test for correctness of expectation 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!'

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 $\mu$.
