2.5 $\chi$2$, t$ and $F$ distribution | Introduction to Statistics and Computation with Data

2.5 $\chi$²$, t$ and $F$ distribution

Exercises

Exercise 2.10 Suppose $X_1, X_2, \dots, X_n$ be $\rm iid$ random sample from a $N(0,1)$ population and $Y_1, Y_2, \dots, Y_{m}$ be another set of $\rm iid$ random sample from a $N(0,1)$ population. Let \[\begin{align*} V = \sum_{i=1}^{m}Y_{i}^2 \qquad \textsf{ and } \qquad U = \sum_{i=1}^{m}X_{i}^2 \end{align*}\]

($\chi^2$ with $n$ degrees of freedom: $\chi_n^2$) Show that $U$ has $\chi_n^2$ distribution, That is the probability distribution function of $U$ is given by, \[\begin{equation} \tag{2.7} f_{U}(u) = \frac{2^{-\frac{n}{2}}}{\Gamma\left( \frac{n}{2} \right)}u^{\frac{n}{2} - 1}e^{-\frac{u}{2}} = \begin{cases} \frac{2^{-\frac{n}{2}}}{\left( \frac{n}{2} - 1 \right)!}u^{\frac{n}{2} - 1}e^{-\frac{u}{2}} & \textsf{ when $n$ is even } \\ \frac{2^{n-\frac{n}{2}-1} \left( \frac{n-1}{2} \right)! }{\left( n - 1 \right)!\sqrt{\pi}}u^{\frac{n}{2} - 1}e^{-\frac{u}{2}} & \textsf{ when $n$ is odd } \end{cases} \qquad \forall u \in \RR \end{equation}\]
($F-$distribution: $F(m,n)$) Let \[Z = \frac{U/n}{V/m}\] Show that $Z$ has $F(n, m)$ distribution, that is its probability density function for $z > 0$ is given by, \[\begin{equation} \tag{2.8} f_{Z}(z) = \left( \frac{m}{n} \right)^{\frac{n}{2}}\frac{z^{\frac{n}{2}-1}}{\left( 1 + \frac{n}{m}z \right)^{\frac{n+m}{2}}}\frac{\Gamma\left( \frac{n+m}{2} \right)}{\Gamma\left( \frac{n}{2} \right)\Gamma\left( \frac{m}{2} \right)} \end{equation}\]
($t_m$-distribution) Let $n=1$, \[W = \sqrt{Z} = \frac{X_1}{\sqrt{\frac{V}{m}}}\] Show that $W$ has $t_{m}$ distribution, that is its probability density function for $w \in \RR$ is given by, \[\begin{equation} \tag{2.9} f_{W}(w) = \frac{\Gamma\left( \frac{m+1}{2} \right)}{\sqrt{m\pi}\Gamma\left( \frac{m}{2} \right)}\left( 1 + \frac{w^2}{m} \right)^{-\frac{m+1}{2}} \end{equation}\]

Exercise 2.11 Distinguishing between Student’s $t-$distribution and Normal distribution:

Using rnorm() and rt() generate $100$ samples of $N(0,1)$ and $t_{25}$ distribution. Compare them using the inbuilt boxplot(), qqnorm() and qqline() functions.
Using range of $[-4.4]$ (in same frame) plot the densities of $N(0,1)$ and $t_{k}$ distributions for $k = 3, 33, 66$ and $99$ using the dnorm(), dt() and plot() functions.

2.5 \(\chi\)2\(, t\) and \(F\) distribution

Exercises

2.5 \(\chi\)²\(, t\) and \(F\) distribution