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5. The t Distribution

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 t Density

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. Let

T = Z / (V / n)1/2.

In the following exercise, you will show that T has probability density function given by

f(t) = C(n) (1 + t2 / n)-(n + 1)/2 for t in R

where the normalizing constant C(n) is given by

C(n) = gam[(n + 1) / 2] / [(npi)1/2 gam(n / 2)].

Mathematical Exercise 1. Show that T has the given density function by using the following steps.

  1. Show first that the conditional distribution of T given V = v is normal with mean 0 and variance n / v.
  2. Use (a) to find the joint distribution of (T, V).
  3. Integrate the joint density in (b) with respect to v to find the density of T.

The distribution of T is known as the Student t distribution with n degree of freedom. The distribution is well defined for any n > 0, but in practice, only positive integer values of n are of interest. This distribution was first studied by William Gosset, who published under the pseudonym Student. In addition to supplying the proof, Exercise 1 provides a good way of thinking of the t distribution: the t distribution arises when the variance of a mean 0 normal distribution is randomized in a certain way.

Simulation Exercise 2. In the random variable experiment, select the student t distribution. Vary n and note the shape of the density function. With n = 5, run the simulation 1000 times with an update frequency of 10. Note the apparent convergence of the empirical density function to the true density function.

Mathematical Exercise 3. 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 converges to infinity and as t converges to -infinity
  4. f(t) is concave upward for t < -an and t > an; f(t) is concave downward for -an < t < an where an = [n / (n + 2)]1/2.

From Exercise 3 it follows that the t distribution is unimodal, with mode at 0.

Mathematical Exercise 4. The t distribution with 1 degree of freedom is known as the Cauchy distribution, named after Augustin Cauchy. Show that the density function is

f(t) = 1 / [pi(1 + t2)], t in R.

The distribution function and the quantile function do not have simple, closed-form representations. Approximate values of these functions can be obtained from the table of the t distribution and from the quantile applet.

Simulation Exercise 5. In the quantile applet, select the student distribution. Vary the parameter and note the shape of the density function and the distribution function. In each of the following cases, find the median, the first and third quartiles, and the interquartile range.

  1. n = 1
  2. n = 2
  3. n = 5
  4. n = 10

Moments

Suppose that T has the t-distribution with n degrees of freedom. The random variable representation in Exercise 1 can be used to find the mean and variance and other moments of T.

Mathematical Exercise 6. Show that

  1. E(T) = 0 if n > 1.
  2. E(T) does not exist if 0 < n <= 1.

In particular, the Cauchy distribution does not have a mean.

Mathematical Exercise 7. Show that

  1. var(T) = n / (n - 2) if n > 2.
  2. var(T) = infinity if 1 < n <= 2.
  3. var(T) does not exist if 0 < n <= 1.

Simulation Exercise 8. In the simulation of the random variable experiment, select the student t distribution. Vary n and note the location and shape of the mean-standard deviation bar. For the following values of n, run the simulation 1000 times with an update frequency of 10. Compare the behavior of the empirical moments with the theoretical results in Exercises 5 and 6.

  1. n = 3.
  2. n = 2.
  3. n = 1.

Mathematical Exercise 9. Show that

  1. E(Tk) = 0 if k is odd and n > k.
  2. E(Tk) = gam[(k + 1) / 2]{gam[(n - k) / 2]}k/2 / [gam(1 / 2)gam(n / 2)] if k is even and n > k.
  3. E(Tk) = infinity if 0 < n <= k.

Normal Approximation

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:

Mathematical Exercise 10. Use a basic limit theorem from calculus to show that for fixed t,

f(t) converges to exp(-t2 / 2) / (2pi)1/2 as n converges to infinity.

Note that the function on the right is the density function of the standard normal distribution.

Mathematical Exercise 11. In the setting of Exercise 1, use the strong law of large numbers to show that, with probability 1,

  1. V / n converges to 1 as n converges to infinity.
  2. T converges to Z as n converges to infinity.

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

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