202405310819
Status: #idea
Tags: Probability Distributions
State: #nascient

Chi-Squared Distribution

Properties

Adding independent ${\chi }^2-$distributed random variables

Let $X$ be distributed according to ${\chi }^2(r_x)$ and $Y$ according to ${\chi }^2(R_Y)$ and let them both be independent such that:

Then:

Prove this using the Moment Generating Functions of ${\chi }^2$.

Dividing independent ${\chi }^2-$distributed random variables

Where $F$ is a Fisher Distribution.

Squaring a $N-$distributed random variable ($N(0,1)$)

When you square a normally distributed random variable with mean $0$ and ${\sigma }^2$ 1, it becomes a

Squaring $N-$distributed independent random variables

From what we now from a Normal Distribution, you can always standardize it (make it $N(0,1)$), by substracting the mean and dividing by the standard deviation.

And by #Squaring a Normal Distribution $N-$distributed random variable ($N(0,1)$) we know that such a random variable squared will have ${\chi }^2$ distribution with $1$ degree of freedom.

Finally, by #Adding independent $ chi 2-$distributed random variables we know that the result will be a ${\chi }^2$ with $r_X+r_Y$ degrees of freedom. Take it all together and you obtain:

For $n$ independent normally distributed random variables, then:

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