[91m[1mTest Failed[22m[39m at [39m[1m/Users/gergaud/git-ENS/Julia-TSE/etudiants/M2/jl_notebook_cell_df34fa98e69747e1a8f8a730347b8e2f_W4sZmlsZQ==.jl:19[22m
Expression: empirique([1.0, 2, 3], 1.5) == 1
Evaluated: 0 == 1
Test.FallbackTestSetException("There was an error during testing")
We are going to illustrate the central limit theorem :
Suppose $X_1,X_2,\ldots$ is a sequence of Independent and identically distributed random variables with $E(X_i)=\mu$ and $Var(X_i)=\sigma^2 < +\infty$. Then, as $n$ approaches infinity, the random variables $\sqrt{n}(\bar{X}_n - \mu)$ converge in distribution to a normal distribution $\mathcal{N}(0,\sigma^2)$
### Exercise
1. Choose a distribution law dist, compute its mean $\mu$ and its variance $\sigma^2$ and $N$ the number of sanple
2. For $n$ in (1,2,5,20)
1. Generate N=10000 samples of lenght n from the dist distribution
2. Compute the means of the $N$ samples and the $N$ values $\sqrt{n}(\bar{X}_n - \mu)$
3. Plot the histogram of these $N$ values and the normal distribution $\mathcal{N}(0,\sigma^2)$