Illustration for other distributions through simulation

Figure 9.1 (Link) illustrates the distribution of ${\overline{X}}_n$ for varying $n$, when each $X_i$ has a Uniform$(0,1)$ distribution. The plots are histograms of $2000$ realisations of ${\overline{X}}_n$ for $n=1$, $10$, $100$ and $1000$, i.e. to make the plot in the lower left-hand corner we have taken the mean of $100$ Uniform$(0,1)$-distributed random variables $2000$ times.

Figure 9.1 (Link) shows that the distribution of ${\overline{X}}_n$ concentrates more and more around $\mu =0.5$ as $n$ gets larger reflecting the fact that the variance decreases to $0$ as $n$ increases. Furthermore the shape of the histogram - even at $n=10$ resembles that of a Normal distribution.

Figure 9.2 (Link) illustrates the distribution of $\sqrt{n}({\overline{X}}_n-\mu )$ for $n=1$, $2$, $5$ and $10$ when the $X$’s are uniform. Figure 9.3 (Link) illustrates the distribution when the $X$’s are exponential.

The pdf for the approximating normal distribution is superimposed on each of the histograms. Note the very fast convergence to a normal in the uniform case and the somewhat slower convergence in the exponential case. In the uniform cases the normal approximation is very good for $n\ge 10$, say. For the exponential it takes to around $n=30$ (exercise: you can plot the density for different $n$ in this case yourself and see the convergence, since $\frac{1}{n}{\sum }_{i=1}^n{X}_i\sim \mathrm{𝖦𝖺𝗆}(n,n)$).