Math 281A Homework 1
Due: Oct 10, in class
1. Suppose Xn∼Binomial(n, pn), and limn→∞npn=λ>0. Show that Xn
d
Ð→ Poisson(λ).
2. Suppose Xn∼χ2
nwith ndegrees of freedom. Find anand bnsuch that
Xn−an
bn
d
Ð→ N(0,1).
3. Let Y1,...,Ynbe i.i.d. samples from Uniform[0,1], and Y(1),...,Y(n)be the order statistics. Show
that n(Y(1),1−Y(n))d
Ð→ (U, V ), where Uand Vare two independent exponential random variables.
4. Let X1,...,Xnbe i.i.d. samples from Uniform[−θ, θ], and X(1),...,X(n)be order statistics. Show
that the following three statistics are asymptotically consistent estimators of θ.
(a) X(n);
(b) −X(1);
(c) (X(n)−X(1))/2.
5. A random variable Xnis said to follow a t-distribution with ndegrees of freedom, if Xn∼√nZ/√Z2
1+...,Z2
n,
where Z, Z1,...,Znare i.i.d. from N(0,1). Show that Xn
d
Ð→ N(0,1).
6. Let X1,...,Xnbe i.i.d. from density fλ,a(x)=λe−λ(x−a), when x≥a, where λ>0 and a∈Rare
unknown parameters. Find the MLE (ˆ
λn,ˆan)of (λ, a), and show that (ˆ
λn,ˆan)P
Ð→ (λ, a).