Suppose $X \sim N(0,1)$ and $W_{n} \sim \chi_{n}^{2}$ independently for any positive integer $n$. Let $V_{n}=X / \sqrt{W_{n}} / n$.

a. We know $V_{n} \sim t_{n}$. Show that $V_{n}^{2}$ follows an $\mathrm{F}$ distribution and specify the parameters.

b. Simulate 1,000 samples of $V_{1}^{2}, V_{8}^{2}$, and $V_{50}^{2}$ using the rt function in $\mathrm{R}$ and plot the histogram for each variable.

c. Identify density functions of $V_{1}^{2}, V_{8}^{2}$, and $V_{50}^{2}$ and use $\mathrm{R}$ to plot the theoretical density curves using $\mathrm{df}($ ) from the histograms from part $b$. foreach variable. Is the shape of the density the exact same as the histogram? Explain.