13.8. In Section 13.5 we have indicated a method by which random samples from a specified distribution may be generated. There are numerous other methods by which this may be done, some of which are to be preferred to the one given, particularly if com- puting devices are available. The following is one such method. Suppose that we want to obtain a random sample from a random variable having a chi-square distribution with 2k degrees of freedom. Proceed as follows: obtain a random sample of size k (with the aid of a table of random numbers) from a random variable which is uniformly dis- tributed over (0, 1), say U1,..., Uk. Then evaluate X = -2 In (U1U2.. Uk) -2 In (U). The random variable X will then have the desired distribution, as we shall indicate below. We then continue this scheme, obtaining another sample of size k from a uniformly distributed random variable, and thus finding the second sample value X2. Note that this procedure requires k observations from a uniformly distributed random variable for every observation from X. To verify the statement made above proceed as follows.

(a) Obtain the moment-generating function of the random variable -2 In (U), where U, is uniformly distributed over (0, 1).

(b) Obtain the moment-generating function of the random variable -2- In (U.), where the U's are independent random variables each with the above distribution. Compare this mgf with that of the chi-square distribution and hence obtain the desired conclusion.