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.
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.
The expected value of a Chi-square random variable is
It can be derived as follows:
The variance of a Chi-square random variable is
It can be derived thanks to the usual variance formula ():
The moment generating function of a Chi-square random variable is defined for any :
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 .
The characteristic function of a Chi-square random variable is
Using the definition of characteristic function, we obtain:
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.
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).
In the following subsections you can find more details about the Chi-square distribution.
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:
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.
Let be a standard normal random variable and let be its square:Then is a Chi-square random variable with 1 degree of freedom.
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.
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.
Below you can find some exercises with explained solutions:
Exercise set 1 (perform calculations involving the Chi-square distribution)
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