45. The proof of the Central Limit Theorem requires calculating the moment generating function for the standardized mean from a random sample of any distribution, and showing that it approaches the moment generating function of the standard normal distribution.

Here we look at a particular case of the Laplace distribution, for which the calculation is simpler.

a. Letting X have pdf , q  x  q, show that MX(t)  1/(1  t2), 1  t  1.

b. Find the moment generating function MY(t) for the standardized mean Y of a random sample from this distribution.

c. Show that the limit of MY(t) is , the moment generating function of a standard normal random variable. [Hint: Notice that the denominator of MY(t) is of the form (1  a/n)n and recall that the limit of this is ea.]