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 . 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.
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 . In particular, the Cauchy distribution does not have a mean. In the simulation of the random variable experiment , select the student t distribution. 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.