Statistics 6401, Winter 2012 1

Limiting Distribution and CLT Problems

1. Let X_{1}, X_{2}, …, X_{n} be a random sample from Exp(λ). Let Y_{1}=min{X_{1}, X_{2}, …, X_{n}}. Find the limiting distribution of Y_{1}.

2. Let X_{1}, X_{2}, …, X_{n} be a random sample from F (x) = 1 − \frac{1}{x}, x ≥ 1.

(a) Find the c.d.f. of Y_{1}=min{X_{1}, X_{2}, …, X_{n}}.

(b) Let Z_{n} = n(1 − F_{X}(Y_{n})) where Y_{n}=max{X_{1}, X_{2}, …, X_{n}}, find

lim_{n→∞} F_{Z_{n}}(t).

3. Let X_{1}, X_{2}, …, X_{n} be independent χ^{2}(1) random variables. Then

Y_{n} = ∑_{i=1}^{n} X_{i} ∼ χ^{2}(n).

The C.L.T. says Z_{n} = \frac{Y_{n}−n}{\sqrt{2n}} ⟶^{d} Z ∼ N(0, 1) .

Show that W_{n} = \sqrt{2Y_{n}} − \sqrt{2n − 1} ⟶^{d} Z ∼ N(0, 1) .