STAT 510 HW #10 – Written
Due Wednesday, November 17, 2006
You can turn it in in class, or to my office (116B IH) or mailbox in 101 IH, by 4PM.
No Mallard this time.
1. Consider the simple linear regression case of Question 4 in Written HW #9, where p= 2,
β= (β0, β1)0, and
X=
1x1
1x2
.
.
..
.
.
1xn
=1nx,
where x= (x1, . . . , xn)0is a vector whose values are not all the same.
(a) Show that
(X0X)−1=1
P(xi−x)2 1
nPx2
i−x
−x1!.
(b) Find the estimate b
β1as a function of
X(yi−y)2,X(xi−x)2,X(yi−y)(xi−x).
[Note that P(xi−x)(yi−y) = Pxiyi−nx y.]
(c) Find the estimate b
β0as a function of b
β1,xand y.
(d) Under what condition are b
β0and b
β1independent?
2. Suppose X1, . . . , Xnare iid Gamma(α, λ), where (α, λ)∈(0,∞)×(0,∞). Find method-
of-moment estimators for αand λ.
3. Suppose that number of telephone calls coming in to a call center follows a Poisson process
with a rate of λcalls per minute, where λ∈(0,∞). That is, if Xis the number of calls
coming in over the course of tminutes, then
X∼P oisson(tλ).
(a) Assuming tis known, what is a reasonable estimate of λbased on X?
(b) Assuming tis known, what is a reasonable estimate of the parameter
θ=Pλ[no calls in the next two minutes]
based on X? (First find θin terms of λ.)
1
FRIDAY
