24. State the central limit theorem.

The cumulative distribution function F of the random variable X is continuous and strictly increasing. Show that Y = F(X) is uniformly distributed. Find the probability density function of the random variable −log(1 − Y ), and calculate its mean and variance.

Let {Xk } be a sequence of independent random variables whose corresponding cumulative distribution functions {Fk } are continuous and strictly increasing. Let Zn = −

1

√n Xn k=1

????

1 + log[1 − Fk (Xk )]

, n = 1, 2, . . . .

Show that, as n → ∞, {Zn} converges in distribution to a normal distribution with mean zero and variance one. (Oxford 2007)