7.12 Let X¯ and S denote the sample mean and standard deviation of the random variables X1, X2,... . Find the limiting distribution of X¯

S and S¯

X in each of the following cases:

1. The random variables are uniform on (0, 1), 2. The random variables are exponential with parameter λ, i.e., their density is:

f(x) = λe−λx1{x>0}

, 3. The random variables are chi-squared with parameter p an integer, that is, f(x) = 1 2p/2Γ(p/2)xp/2e−x/21{x>0}, and Γ(t) = ∞

0 xt−1e−xdx denotes the Gamma function.