# Chi-square distribution

A random variable has a Chi-square distribution if it can be written as a sum of squares of independent standard normal variables.

Sums of this kind are encountered very often in statistics, especially in the estimation of variance and in hypothesis testing.

In this lecture, we derive the formulae for the mean, the variance and other characteristics of the chi-square distribution.

## Degrees of freedom

We will prove below that a random variable has a Chi-square distribution if it can be written aswhere , ..., are mutually independent standard normal random variables.

The number of variables is the only parameter of the distribution, called the degrees of freedom parameter. It determines both the mean (equal to ) and the variance (equal to ).

## Definition

Chi-square random variables are characterized as follows.

Definition Let be a 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 and only if its probability density function iswhere is a constant:and is the Gamma function.

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

## Symbol

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".

## Expected value

The expected value of a Chi-square random variable is

Proof

It can be derived as follows:

The proof above uses the probability density function of the distribution. An alternative, simpler proof exploits the representation (demonstrated below) of as a sum of squared normal variables.

Proof

We can writewhere are independent standard normal variables. Then, we havebecause a standard normal variable has zero mean and unit variance.

## Variance

The variance of a Chi-square random variable is

Proof

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

Again, there is also a simpler proof based on the representation (demonstrated below) of as a sum of squared normal variables.

Proof

We can writewhere are independent standard normal variables. Then, we havebecause a standard normal variable has zero mean, unit variance and fourth moment equal to .

## 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 obtainThe 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.

## Density plots

This section shows the plots of the densities of some Chi-square random variables. These plots help us to understand how the shape of the Chi-square distribution changes by changing the degrees of freedom parameter.

### Plot 1 - Increasing the degrees of freedom

The following plot contains the graphs of two density functions:

• the first graph (red line) is the probability density function of a Chi-square random variable with degrees of freedom;

• the second graph (blue line) is the probability density function of a Chi-square random variable with degrees of freedom.

The thin vertical lines indicate the means of the two distributions. By increasing the number of degrees of freedom, we increase the mean of the distribution, as well as the probability density of larger values.

### Plot 2 - Increasing the degrees of freedom

The following plot also contains the graphs of two density functions:

• the first graph (red line) is the probability density function of a Chi-square random variable with degrees of freedom;

• the second graph (blue line) is the probability density function of a Chi-square random variable with degrees of freedom.

As in the previous plot, the mean of the distribution increases as the degrees of freedom are increased.

## Solved exercises

Below you can find some exercises with explained solutions.

### Exercise 1

Let be a chi-square random variable with degrees of freedom.

Compute the following probability:

Solution

First of all, we need to express the above probability in terms of the distribution function of :where the valuescan be computed with a computer algorithm or found in a Chi-square distribution table (see the lecture entitled Chi-square distribution values).

### Exercise 2

Let and be two independent normal random variables having mean and variance .

Compute the following probability:

Solution

First of all, the two variables and can be written aswhere and are two standard normal random variables. Thus, we can writebut the sum has a Chi-square distribution with degrees of freedom. Therefore,where is the distribution function of a Chi-square random variable with degrees of freedom, evaluated at the point . With any computer package for statistics, we can find

### Exercise 3

Suppose that the random variable has a Chi-square distribution with degrees of freedom.

Define the random variable as follows:

Compute the expected value of .

Solution

The expected value of can be easily calculated using the moment generating function of :Now, by exploiting the linearity of the expected value, we obtain

## How to cite

Taboga, Marco (2021). "Chi-square distribution", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/probability-distributions/chi-square-distribution.

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