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# Chi-square distribution

A random variable has a Chi-square distribution if it can be written as a sum of squares:where , ..., are mutually independent standard normal random variables. The importance of the Chi-square distribution stems from the fact that sums of this kind are encountered very often in statistics, especially in the estimation of variance and in hypothesis testing.

## Definition

Chi-square random variables are characterized as follows.

Definition Let be an absolutely continuous random variable. Let its support be the set of positive real numbers:Let . We say that has a Chi-square distribution with degrees of freedom if its probability density function iswhere is a constant:and is the Gamma function.

The following notation is often employed to indicate that a random variable has a Chi-square distribution with degrees of freedom:where the symbol means "is distributed as".

To better understand the Chi-square distribution, you can have a look at its density plots.

## Expected value

The expected value of a Chi-square random variable is

Proof

It can be derived as follows:

## Variance

The variance of a Chi-square random variable is

Proof

It can be derived thanks to the usual variance formula ():

## Moment generating function

The moment generating function of a Chi-square random variable is defined for any :

Proof

Using the definition of moment generating function, we obtain:The integral above is well-defined and finite only when , i.e., when . Thus, the moment generating function of a Chi-square random variable exists for any .

## Characteristic function

The characteristic function of a Chi-square random variable is

Proof

Using the definition of characteristic function, we obtain:

## Distribution function

The distribution function of a Chi-square random variable iswhere the functionis called lower incomplete Gamma function and is usually computed by means of specialized computer algorithms.

Proof

This is proved as follows:

Usually, it is possible to resort to computer algorithms that directly compute the values of . For example, the MATLAB command

`chi2cdf(x,n)`

returns the value at the point `x` of the distribution function of a Chi-square random variable with `n` degrees of freedom.

In the past, when computers were not widely available, people used to look up the values of in Chi-square distribution tables, where is tabulated for several values of and (see the lecture entitled Chi-square distribution values).

## More details

In the following subsections you can find more details about the Chi-square distribution.

### The sum of independent chi-square random variables is a Chi-square random variable

Let be a Chi-square random variable with degrees of freedom and another Chi-square random variable with degrees of freedom. If and are independent, then their sum has a Chi-square distribution with degrees of freedom:This can be generalized to sums of more than two Chi-square random variables, provided they are mutually independent:

Proof

This can be easily proved using moment generating functions. The moment generating function of isDefineThe moment generating function of a sum of mutually independent random variables is just the product of their moment generating functions:where Therefore, the moment generating function of is the moment generating function of a Chi-square random variable with degrees of freedom, and, as a consequence, is a Chi-square random variable with degrees of freedom.

### The square of a standard normal random variable is a Chi-square random variable

Let be a standard normal random variable and let be its square:Then is a Chi-square random variable with 1 degree of freedom.

Proof

For , the distribution function of iswhere is the probability density function of a standard normal random variable:For , because , being a square, cannot be negative. Using Leibniz integral rule and the fact that the density function is the derivative of the distribution function, the probability density function of , denoted by , is obtained as follows (for ):For , trivially, . As a consequence,Therefore, is the probability density function of a Chi-square random variable with 1 degree of freedom.

### The sum of squares of independent standard normal random variables is a Chi-square random variable

Combining the two facts above, one trivially obtains that the sum of squares of independent standard normal random variables is a Chi-square random variable with degrees of freedom.

## Solved exercises

Below you can find some exercises with explained solutions:

1. Exercise set 1 (perform calculations involving the Chi-square distribution)

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