Distributions derived from normal distribution

$χ^2, t,$ and $F$ Distributions

we know that the sum of independent gamma random variables that have the same value of $λ$ follows a gamma distribution, and therefore the chi-square distribution with $n$ degrees of freedom is a gamma distribution with $α = n/2$ and $λ = 1/2$ . Its density is

From the density function of Proposition A, $f (t) = f (−t)$, so the $t$ distribution is symmetric about zero. As the number of degrees of freedom approaches infinity, the $t$ distribution tends to the standard normal distribution; in fact, for more than 20 or 30 degrees of freedom, the distributions are very close. Figure below shows several $t$ densities. Note that the tails become lighter as the degrees of freedom increase.

It can be shown that, for $n > 2, E(W)$ exists and equals $n/(n − 2)$. From the definitions of the $t$ and $F$ distributions, it follows that the square of a $t_n$ random variable follows an $F_{1,n}$ distribution.

The Sample Mean and the Sample Variance

Let $X_1, . . . , X_n$ be independent $N(μ, σ^2)$ random variables; we sometimes refer to them as a sample from a normal distribution. In this section, we will find the joint and marginal distributions of

$\bar{X}=\frac{1}{n} \sum_{i=1}^n X_i$

$S^2 = \frac{1}{n-1}  \sum_{i=1}^n (X_i-X)^2$

These are called the sample mean and the sample variance, respectively. First note that because $X$ is a linear combination of independent normal random variables, it is normally distributed with

$E(X) = μ$
$Var(X) = σ^2/n$

As a preliminary to showing that $\bar{X}$ and $S^2$ are independently distributed, we establish the following theorem.