Statistics Ph.D.
Qualifying Exam: Part I
October 27, 2001
Student Name:
1. Answer 8 out of 12 problems. Mark the problems you selected
in the following table.
1 2 3 4 5 6 7 8 9 10 11 12
2. Write your answer right after each problem selected, attach more
pages if necessary.
3. Assemble your work in right order and in the original problem
order.
1. Suppose that X has a uniform distribution on the unit interval (0, 1). Given X, the
random variable Y is uniform on the interval (0, X).
(a) Find E(X), V ar(X), E(Y ), and V ar(Y ).
(b) Find E(X|Y ) and E(Y |X).
(c) Find V ar(X|Y ) and V ar(Y |X).
2. Suppose that X1 , X2 and X3 have a trinomial distribution with index n and probability
P
parameters p1 , p2 and p3 , where pj = 1. The log likelihood function is
X
l(p1 , p2 , p3 ) = Xj log pj ,
and the observed values of the Xj ’s are 32, 46 and 22.
(a) Find the maximum likelihood estimates of pj ’s.
(b) Find the maximum likelihood estimates of pj ’s, when the pj ’s satisfy the hypoth-
esis
p1 = θ2 , p2 = 2θ(1 − θ), p3 = (1 − θ)2 .
3. Suppose that the joint p.d.f. of two random variables X and Y is f (x, y) = x+y, 0≤
x, y ≤ 1.
(a) Find P (2X + Y ≤ 1).
(b) Find the p.d.f. of Z = XY .
4. Let X, Y, U be independent random variables with X ∼ Poisson(λ), Y ∼ Poisson(µ),
and U ∼ Uniform(0, 1). Let
(
X , if U > a,
V =
Y , if U ≤ a,
where a is a constant in (0, 1). Find
P (X + Y = n | V = k).
5. Let X1 , . . . , Xn be a random sample of size n from the p.d.f.
f (x; θ) = θx−2 , 0 < θ ≤ x < ∞.
(a) Find the MLE of θ.
(b) Find the method of moment estimator of θ based on E(X 1/2 ).
(c) Find the method of moment estimator of θ based on E(X −1 ).
6. Imagine a population of N + 1 boxes. Box number k contains k red and N − k green
balls (k = 0, 1, . . . , N ). A box is chosen at random and n random drawings are made
from it, the ball drawn being replaced each time by a ball with the opposite color.
Define
Event A: All n balls turn out to be red,
Event B: The (n + 1)st draw yields a red ball.
(a) Find P (A|Box k is chosen) (k = 0, 1, . . . , N ).
(b) Find P (A).
(c) Find P (A ∩ B).
(d) Find P (B|A).
(e) Find an approximation to P (B|A), using the fact that if N is large,
N Z 1
1 X k 1
( )n ∼ xn dx = .
N k=1 N 0 n+1
7. Let X1 , . . . , Xn be i.i.d. from the uniform distribution on the interval (θ, θ + 1).
(a) Find the joint distribution of X(1) and X(n) , where X(1) = min(X1 , . . . , Xn ), and
X(n) = max(X1 , . . . , Xn ).
(b) Find a UMP test of size α for testing H0 : θ ≤ 0 versus H1 : θ > 0.
8. Let A, B, C be a random sample of size 3 from U (0, 1) distribution. Find the probability
that
Ax2 + Bx + C = 0
has real roots.
9. Let X1 , . . . , Xn be i.i.d. from the density f (x; θ) = 3x2 /θ3 if 0 < x < θ and f (x; θ) = 0
otherwise.
(a) Show that X(n) = max(X1 , . . . , Xn ) is a sufficient and complete statistic for θ.
(b) Find the UMVUE for θ.
10. Let {X1 , . . . , Xn } be a random sample from the population with density f (x, θ) =
θ xθ−1 , 0 < x < 1, θ > 0.
(a) Derive the UMP (Uniformly Most Powerful) size α test for testing H0 : θ = 1
versus H1 : θ > 1.
(b) What is the power function of your test ?
(c) Show that UMP size-α test for testing H0 : θ = 1 versus H2 : θ 6= 1 does not
exist.
11. Let X1 , . . . , Xm and Y1 , . . . , Yn be independent samples from Poisson(λµ) and Poisson(µ)
populations respectively.
(a) Find the MLE’s (λ̂, µ̂) for (λ, µ).
(b) Find jointly minimal sufficient statistics S for (λ, µ).
(c) Is (λ̂, µ̂) a function of S?
(d) Is (λ̂, µ̂) jointly sufficient for (λ, µ)?
12. The normally distributed random variables X1 , . . . , Xn are said to be serially correlated
or to follow an autoregressive model if we can write
Xi = θXi−1 + i , i = 1, . . . , n,
where X0 = 0 and 1 , . . . , n are independent N (0, σ 2 ) random variables.
0
(a) Show that the density of X = (X1 , . . . , Xn ) is
n
2 −n/2 2
(xi − θxi−1 )2 },
X
p(x, θ) = (2πσ ) exp { − (1/2σ )
i=1
for −∞ < xi < ∞, i = 1, . . . , n and x0 = 0.
(b) Show that the likelihood ratio statistic of H0 : θ = 0 (independence) vs H1 : θ 6= 0
(serial correlation) is equivalent to
n n−1
Xi Xi−1 )2 / Xi2 .
X X
−(
i=2 i=1
