What is an example of Chi-Square Distribution?

The chi-square distribution is a probability density distribution consisting of one parameter, degrees of freedom. Some of the other distributions to which it relates are:

[sqrt(Y~chi(d)]^2 ~ chi-squared(d)

Sum from i = 1 to n of {Z(i)^2} ~ chi-squared(n) where Z(i) ~ N(0, 1) for all i

That is, the sum of the squares of n i.i.d standard normal random variables is chi-squared distributed with n degrees of freedom.

[d1 * (F ~ F(d1, d2)] ~ chi-squared(d1) as d2 → ∞

And probably the most commonly used identity is that two chi-squared random variables with d1 and d2 degrees of freedom respectively, divided by their degrees of freedom, is F-distributed with d1 and d2 degrees of freedom. That is:

{[X ~ chi-squared(d1)] / d1} / {[Y ~ chi-squared(d2)] / d2} ~ F(d1, d2)

Exp(0.5) = chi-squared(2) where the parameter of the exponential is the rate parameter, not scale.

From this last identity, an example of the chi-square distribution is that whenever you have an exponential distribution with rate parameter 0.5, it is equivalent to the chi-squared distribution with parameter 2.

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