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  2. 4. Special Distributions
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5. The Student \(t\)

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 Probability Density Function

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

\[ T = \frac{Z}{\sqrt{V / n}} \]

\(T\) has probability density function \(f\) given by

\[ f(t) = \frac{\Gamma[(n + 1) / 2]}{\sqrt{n \, \pi} \, \Gamma(n / 2)} \left( 1 + \frac{t^2}{n} \right)^{-(n + 1) / 2}, \quad t \in \R \]
Proof:

The conditional distribution of \(T\) given \(V = v\) is normal with mean 0 and variance \(n / v\). Use this fact to find the joint probability density function of \((T, V)\). Then integrate the joint probability density function in (b) with respect to \(v\) to find the probability density function 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 \(n \gt 0\), 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.

In the special distribution simulator, select the student \(t\) distribution. Vary \(n\) and note the shape of the probability density function. For selected values of \(n\), run the simulation 1000 times and note the apparent convergence of the empirical density function to the true probability density function.

The Student probability density function has the following properties:

  1. \(f\) is symmetric about \(t = 0\).
  2. \(f\) is increasing on \((-\infty, 0)\) and decreasing on \((0, \infty)\).
  3. The maximum of \(f\) occurs at \(t = 0\).
  4. \(f\) is concave upward on \((-\infty, a_n)\) and on \((a_n, \infty)\); \(f\) is concave downward on \((-a_n, a_n)\), where \(a_n = \sqrt{n / (n + 1)}\).
  5. The inflection points occur at \(t = \pm a_n\).
  6. \(f(t) \to 0\) as \(t \to \infty\) and as \(t \to -\infty\).
  7. \(a_n \to 1\) as \(n \to \infty\).

In particular, if follows that the distribution is unimodal with mode and median at \(t = 0\).

The \(t\) distribution with 1 degree of freedom is known as the Cauchy distribution, named after Augustin Cauchy. The probability density function is

\[ f(t) = \frac{1}{\pi \, (1 + t^2)}, \quad t \in \R \]

The probability density function of the Cauchy distribution can be obtained by normalizing the function

\[ g(t) = \frac{1}{1 + t^2}, \quad t \in \R \]

We recognize \(g\), of course, as the derivative of the arctangent function. In addition, the graph of \(g\) is also known as the witch of Agnesi, named for the Italian mathematician Maria Agnesi.

The distribution function and the quantile function of the general \(t\) distribution do not have simple, closed-form representations. Approximate values of these functions can be obtained from the special distribution calculator, and from most mathematical and statistical software packages. However, we can find simple formulas in the special case of the Cauchy distribution.

Let \(F\) denote the distribution function of the Cauchy distribution. Then

  1. \(F(t) = \frac{1}{\pi} \, \arctan(t) + \frac{1}{2}\) for \(t \in \R\)
  2. \(F^{-1}(t) = \tan[\pi(p - \frac{1}{2})]\) for \(p \in (0, 1)\)
  3. The first quartile is \(t = -1\) and the third quartile is \(t = 1\).

In the special distribution calculator, 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. \(n = 2\)
  2. \(n = 5\)
  3. \(n = 10\)
  4. \(n = 20\)

Moments

Suppose that \(T\) has a \(t\) distribution. The basic random variable representation that we started with can be used to find the mean and variance and other moments of \(T\).

If \(T\) has the \(t\) distribution with \(n\) degrees of freedom then

  1. \(\E(T)\) is undefined if \(0 \lt n \le 1\)
  2. \(\E(T) = 0\) if \(1 \lt n \lt \infty\)

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

If \(T\) has the \(t\) distribution with \(n\) degrees of freedom then

  1. \(\var(T)\) is undefined if \(0 \lt n \le 1\)
  2. \(\var(T) = \infty\) if \(1 \lt n \le 2\)
  3. \(\var(T) = \frac{n}{n - 2}\) if \(2 \lt n \lt \infty\)

Note that \(\var(T) \to 1\) as \(n \to \infty\).

In the simulation of the special distribution simulator, 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. Compare the behavior of the empirical moments with the theoretical results in Exercise 7 and Exercise 8.

  1. \(n = 3\)
  2. \(n = 2\)
  3. \(n = 1\)

If \(T\) has the \(t\) distribution with \(n\) degrees of freedom, then

  1. \(\E(T^k)\) is undefined if \(k\) is odd and \(k \ge n\)
  2. \(\E(T^k) = \infty\) if \(k\) is even and \(k \ge n\)
  3. \(\E(T^k) = 0\) if \(k\) is odd and \(k \lt n\)
  4. Finally, if \(k\) is even and \(k \lt n\) then \[ \E(T^k) = \frac{\Gamma[(k + 1) / 2] \, \Gamma^{k/2}[(n - k) / 2]}{\sqrt{\pi} \, \Gamma(n / 2} \]

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:

For fixed \(t\),

\[ f(t) \to \frac{1}{\sqrt{2 \, \pi}} e^{-\frac{1}{2} t^2} \text{ as } n \to \infty \]
Proof:

Use a basic limit theorem from calculus.

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

In the basic random variable representation, \(T \to Z\) as \(n \to \infty\) with probability 1.

Proof:

Note that \(V / n \to 1\) as \(n \to \infty\) with probability 1 by the strong law of large numbers.

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

Suppose that \(T\) has the \(t\) distribution with \(n = 10\) degrees of freedom. For each of the following, compute the true value using the special distribution calculator and then compute the normal approximation. Compare the results.

  1. \(\P(-0.8 \lt T \lt 1.2)\)
  2. The 90th percentile of \(T\).
  1. Virtual Laboratories
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