## The Chi-Square Distribution |

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.

** 1. **Suppose that random variable *Z *has the standard normal distribution. Use change
of variable techniques to show that *U* = Z^{2}
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 *Z*_{1},
*Z*_{2}, ..., *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.

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.

** 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

*f*(*v*) decreases as*v*increases.*f*(*v*) increases to infinity as*v*decreases to 0.*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

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

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

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

** 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

** 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 |