Question

Please give me the matlab codes, and capture the generated figures.

$Screenshot_20210419-233033_Office.jpg$

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Fig.

4.6. Quantile-quantile plot for samples of the form (4.7) against \(\mathrm{N}(0,1)\) quantiles.

Computational example In Figures \(4.5\) and \(4.6\) we use the techniques introduced above to show the remarkable power of the Central Limit Theorem. Here, we generated sets of \(\mathrm{U}(0,1)\) samples \(\left\{\xi_{i}\right\}_{i=1}^{n}\), with \(n=10^{3}\). These were combined to give samples of the form

$$ \frac{\sum_{i=1}^{n} \xi_{i}-n \mu}{\sigma \sqrt{n}} $$

where \(\mu=\frac{1}{2}\) and \(\sigma^{2}=\frac{1}{12} .\) We repeated this \(M=10^{4}\) times. These \(M\) data points were then used to obtain a kernel density estimate. In Figure \(4.5\) we used bins of width \(\Delta x=0.5\) over \([-4,4]\) and plotted \(N_{i} /(M \Delta x)\) against \(x_{i}\), as described for Figure 4.1. Here we have used a histogram, or bar graph, so each rectangle is centred at an \(x_{i}\) and has height \(N_{i} /(M \Delta x)\). The \(\mathrm{N}(0,1)\) density curve is superimposed as a dashed line. Figure \(4.6\) gives the corresponding quantile-quantile plot. The figures confirm that even though each \(\xi_{i}\) is nothing like normal, the scaled sum \(\left(\sum_{i=1}^{n} \xi_{i}-n \mu\right) /(\sigma \sqrt{n})\) is very close to \(\mathrm{N}(0,1) . \quad \diamond\)