The Student t Distribution

Java Applet Simulation of the Student t 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 a standardized version of the sample mean when the underlying distribution is normal.

The Student t Density

1. Suppose that Z has the standard normal distribution, V has the chi-squared distribution with n degrees of freedom and that Z and V are independent. Show that the probability density function of

is given by

The distribution defined by the density function in Exercise 3 is known as the Student t distribution with n degrees of freedom. This distribution was first studied by William Gosset, who published under the pseudonym Student.

Simulation

The applet for this page simulates a random variable with a t 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 from the 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.

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

3. 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.

Properties

4. Sketch the graph of the t density function given in Exercise 1. In particular, show that

1. f(t) is symmetric about t = 0.
2. f(t) increases for t < 0 and decreases for t > 0
3. f(t) converges to 0 as t approaches infinity or negative infinity.

You probably noticed that, qualitatively at least, the t density function is very similar to the standard normal density function. The similarity is quantitative as well:

5. Use a basic limit theorem from calculus to show that for fixed t,

f(t) converges to j(t) as n increases to infinity

where j is the density function of the standard normal distribution.

The t distribution has more probability in the tails, and consequently less probability near 0, compared to the standard normal distribution.

6. Show that the t distribution with 1 degree of freedom is the same as the Cauchy distribution.

7. Use symmetry to show that when n > 1, the t distribution with n degrees of freedom has mean 0.

8. Use the random variable representation in Exercise 4 to show that when n > 2, the t distribution with n degrees of freedom has variance n / (n - 2)

Special Distributions