Definition – Chi-Square distribution.

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1. Definition

Definition

Recall the definition of the normal distribution,

$p(Z=x) = \frac{1}{\sigma\sqrt{2\pi }} e^{-\frac{1}{2} (\frac{z-\mu}{\sigma})^2}$

We define the chi-square distribution to be the distribution of the sum of $n\in\mathbb{N}$ iid standard normal variables.

That is,

$\sum_{i=0}^n Z_i = Z_1 + ... + Z_n = Y$

follows a $\chi_n^2$ distribution. The pdf of the Chi-square distribution is given by

$p_Y(y) = \frac{1}{2^{\frac{n}{2}}\Gamma(\frac{n}{2})}y^{\frac{n}{2}-1}e^{-\frac{1}{2} y}$

Derivation

Now we have defined the distribution, we now need its pdf. This derivation we will make use of the gamma distribution, and the fact that for $X_1 \sim Gamma(\alpha_1,\beta)$ and $X_2 \sim Gamma(\alpha_2,\beta)$ ,

$X_1 + X_2 \sim Gamma(\alpha_1 + \alpha_2,\beta)$ .

Let $Z \sim N(0,1)$ . Then, for a bijective function $g(Z)=Z^2=Y$ , the pdf $p_Y(y)$ can be written as

$p_Y(y) = p_X(g^{-1}(y)) |\frac{dg^{-1}(y)}{dy}|$ .

(This is true for any distribution). For the the case of $n=1$ , we have that if $Z\sim N(0,1)$ , then as $g(Z) = Z^2$ is bijective for $x>0$ , and we have via the above formula,

$p_Y(y) = p_X(\sqrt{y}) |\frac{-1}{2\sqrt{y}}|$ .

Inserting $\sqrt{y}$ into the pdf for $Z$ , we have, that

$p_Y(y) = \frac{1}{\sqrt{2\pi }} e^{-\frac{\sqrt{y^2}}{2}} |\frac{-1}{2\sqrt{y}}| \\\\ = \frac{1}{\sqrt{2\pi y}} e^{-\frac{y}{2}} \frac{1}{2}$ .

But for $x<0$ , we obtain the exact same function. Thus, for each $g(z) = z^2 = y$ , we have that either $z = \pm y$ . But as the normal distribution is symmetrical around $0$ , we have that their probabilities are equal, so multiplying $p_Y(y)$ by two gives

$p_Y(y) = \frac{1}{\sqrt{2\pi y}} e^{-\frac{y}{2}} \quad (1)$

Now recall the pdf for the gamma distribution,

$p(X=x) = \frac{\beta^\alpha}{\Gamma (\alpha)}x^{\alpha -1}e^{-\beta x}$ .

Comparing terms, we can see that $(1)$ is a gamma distribution, $Gamma(\frac{1}{2},\frac{1}{2})$ .

Finally, to complete the derivation, recall that for $n$ gamma distributed, iid, random variables $\{X_1, ..., X_n \}$ , with each $X_i \sim Gamma(\alpha,\beta)$ ,

$\sum_{i=1}^n X_i \sim Gamma(n\alpha,\beta)$ .

Thus, because the pdf $(1)$ is a gamma distribution, for the sum of $n$ standard normal variables, we need simply replace the first parameter with $\frac{n}{2}$ rather than $\frac{1}{2}$ . Doing so yields the pdf for the Chi-square distribution,

$\frac{ (\frac{1}{2})^\frac{n}{2}}{\Gamma (\frac{n}{2})}x^{\frac{n}{2}-1}e^{-\frac{1}{2} x} = \frac{1}{2^{\frac{n}{2}}\Gamma(\frac{n}{2})}x^{\frac{n}{2}-1}e^{-\frac{1}{2} x}$ .

Thus, the Chi-square pdf is

$\frac{1}{2^{\frac{n}{2}}\Gamma(\frac{n}{2})}x^{\frac{n}{2}-1}e^{-\frac{1}{2} x}$