The Chi-Square Distribution

Java Applet Simulation of the chi-square experiment

In this section we will study a distribution that has special importance in statistics. In particular, this distribution will arise in the study of the sample variance when the underlying distribution is normal.

The Chi-Square Density

1. Suppose that random variable Z has the standard normal distribution. Use change of variable techniques to show that U = Z² has probability density function

2. Show that the density function in Exercise 1 is the gamma density with parameters 1/2 and 1/2.

3. Suppose that Z₁, Z₂, ..., Z_n are independent standard normal variables. Use the result of Exercise 2 and properties of the gamma distribution to show that

has the gamma distribution with parameters n / 2 and 1 / 2 and hence has density function

The special gamma distribution in Exercise 3 is called the chi-square distribution with n degrees of freedom. In particular, the distribution in Exercise 1 is the chi-square distribution with 1 degree of freedom.

Simulation

The applet for this page simulates a random variable with the chi-square distribution. The graph of the density function is shown in blue. The mean and standard deviation are recorded in the first table and shown graphically in terms of the horizontal blue bar, which is centered at the mean and extends one standard deviation on either side. You can vary the degrees of freedom parameter with a scroll bar. When you run the experiment, independent values of the chi-square distribution are generated and recorded in the second table. The empirical density function is graphed in red and the sample mean and variance recorded in the first table and shown graphically in terms of the horizontal red bar.

4. Start the applet and vary the degrees of freedom with the scroll bar. Note the shape of the chi-square density function and the location and size of the mean-standard deviation bar.

5. Run the simulation with an update frequency of 100 for several different degrees of freedom. Note the apparent convergence of the empirical density and statistics to the corresponding distribution density and parameters.

Basic Properties

6. Show that the chi-square distribution with 2 degrees of freedom is the same as the exponential distribution with parameter 1/2.

7. Sketch the graph of the chi-square density function with n = 1 degrees of freedom. In particular, show that

1. f(v) decreases as v increases.
2. f(v) increases to infinity as v decreases to 0.
3. f(v) decreases to 0 as v increases to infinity.

8. Sketch the graph of the chi-square distribution with n = 2 degrees of freedom. In particular, show that

1. f(v) decreases as v increases.
2. f(v) increases to 1/2 as v decreases to 0.
3. f(v) decreases to 0 as v increases to infinity.

9. Sketch the graph of the chi-square distribution with n > 2 degrees of freedom. In particular, show that

1. f(v) increases for 0 < v < n - 2 and decreases for v > n - 2.
2. f(v) decreases to 0 as v decreases to 0.
3. f(v) decreases to 0 as v increases to infinity.

From Exercise 9 (a) it follows that the mode of the distribution occurs at n - 2.

Moments

10. Use properties of the gamma distribution to show that the chi-square distribution with n degrees of freedom has mean n and variance 2n.

11. Suppose that V has the chi-square distribution with n degrees of freedom. Use properties of the gamma distribution to show that the moment generating function is

Transformation

12. Use Exercise 11 to show that if U has the chi-square distribution with m degrees of freedom, V has the chi-square distribution with n degrees of freedom, and U and V are independent, then U + V has the chi-square distribution with m + n degrees of freedom.

Special Distributions