2008 Probability / Mathematical Statistics Comprehensive Exam (100 Points)
1. Suppose P (A0 ) = .4, P (B 0 ) = .7, and P (A ∩ B) = .2. Find:
(a) (2pt) P (A ∪ B)
(b) (3pt) P (B|A)
(c) (3pt) P (B|A0 )
(d) (2pt) Are events A and B independent? Justify your answer.
(
2. X is a continuous random variable with pdf f (x) =
(x − 1)/4 if 2 ≤ x ≤ 4
0
otherwise
(a) (4pt) Find E(X).
(b) (4pt) Find the CDF F (x). Be sure to define F (x) for all real numbers.
3. (3pt) In a box there are 10 identical balls except for the colors: 4 are white and 6 are red. If I
pick a white ball I win a prize. If I pick a red ball, I pay $2.00. The ball is replaced after being
selected. Let X be the number of times I play until (and including) the second time I win. What
is the distribution of X and what are the associated parameter values?
4. (3pt) In a manufacturing process that makes light bulbs, there is a 0.01 probability that any
particular bulb produced is defective. Assume this probability is constant over time. The bulbs
are put into boxes containing 100 bulbs. Let X be the number of defective bulbs in a randomly
selected box of bulbs. What is the distribution of X and what are the associated parameter
values?
5. (4pt) X and Y are random variables with joint pdf f (x, y) =

 1
 2
xy
for 0 < y < x < 2
0
otherwise
Find fX (x), the marginal pdf for X.
6. Two discrete random variables X and Y have joint pdf defined by the following table:
Y
X
1
2
1
.10
.30
2
.15
.45
(a) (2pt) What is the marginal pdf fY (y)?
(b) (2pt) What is the conditional pdf f (Y |X = 1)?
(c) (2pt) What is E(Y |X = 1)?
(d) (2pt) Are X and Y independent? Justify your answer.
7. (6pt) Find the pdf of Z =
fX (x) =

2




x3



 0
1
if X is a random variables with pdf and CDF
X
for 1 < x < ∞
FX (x) =
otherwise

1



 1− 2
for 1 < x < ∞




otherwise
x
0
8. Suppose X1 ∼ EXP(3), X2 ∼ GAMMA(3, 4), and X1 and X2 are independent.
(a) (5pt)
What is the moment generating function of Y = X1 + X2 ?
(b) (3pt)
What is the distribution of Y ?
9. (8pt) Verify that X is a UMVUE of µ for a Poisson(µ) distribution based on a random sample
of size n.
10. Let X1 , X2 , . . . , Xn be a random sample from f (x; θ) = θxθ−1 with support 0 < x < 1 and with
θ > 0. Note this is the Beta(θ, 1) pdf.
(a) (3pt) Find the method of moments estimator (MME) of θ.
b the maximum likelihood estimator (MLE) of θ.
(b) (6pt) Find θ,
1
(c) (2pt) What is the MLE of
?
θ+1
Suppose X and Y are independent EXP(1) random variables. Find the joint pdf of
11. (8pt)
S = X + Y and T = X.
12. (5pt) Find a complete sufficient statistic if we have a random sample X1 , X2 , . . . , Xn from a
(ln θ)θx
distribution with pdff (x; θ) =
for 0 < x < 1 and θ > 1.
θ−1
13. (7pt) Let X1 , X2 , . . . , Xn be a random sample from a Gamma(2, 1/θ) pdf. It can be shown that
2θXi ∼ χ2 (4). Determine an equal-tailed 100γ% interval estimator for θ based on Y =
n
X
2θXi .
i=1
14. (5pt) Suppose X1 , . . . , X20 are a random sample from an EXP(2) distribution. Approximate
the probability that X ≥ 2.5.
15. (6pt) Suppose X and Y are a random sample from an EXP(1) distribution. Find the joint pdf
of S = X + Y and T = X.
Probability / Math Stat Handout to Accompany Exam
• If T is an unbiased estimator of τ (θ), then the Cramer-Rao lower bound (CRLB) (given
a random sample) is
[τ 0 (θ)]2
Var(T ) ≥
h
i2
∂
nE ∂θ
ln f (X; θ)
assuming (i) the derivatives exist and (ii) they can be passed into the integral (continuous X)
or summation (discrete X) when calculating expectations.
• Let statistic Si = Si (X1 , . . . , Xn ) and and si = Si (x1 , . . . , xn ) for i = 1, . . . , k. Let S(x1 , . . . , xn )
be the vector of functions whose j th coordinate is Sj (x1 , . . . , xn ) Then, mathematically, if the
conditional pdf of X given S = s,
fX|s (x1 , x2 , . . . , xn ) =
f (x1 , x2 , . . . , xn ; θ)
fS (s; θ)
if S(x1 , . . . , xn ) = s
and 0 otherwise
does not depend on θ, then S is a set of sufficient statistics.
• Theorem 10.2.1 (Factorization Criterion): If X1 , . . . , Xn is a random sample having
joint pdf f (x1 , . . . , xn ; θ), and if S = (S1 , . . . , Sk ), then S is jointly sufficient for θ if and only
if
f (x1 , . . . , xn ; θ) = g(s; θ) h(x1 , . . . , xn )
where g(s; θ) does not depend on observations x1 , . . . , xn except through s and h(x1 , . . . , xn )
does not involve θ.
• Theorem 10.3.1: If S1 , . . . , Sk are jointly sufficient for θ and if θb is a unique MLE of θ,
then θb is a function of S = (S1 , . . . , Sk ).
• A density function is a member of the regular exponential class (REC) if it can be
expressed in the form
f (x; θ) = c(θ)h(x) exp
" k
X
#
qj (θ)tj (x)
for x ∈ A (support)
i=1
and is 0 otherwise, where θ = (θ1 , . . . , θk ) is a vector of unknown parameters and the parameter
space Ω has the form
Ω = {θ| ai < θi < bi , i = 1, . . . , k}
(Note: ai = −∞ and bi = ∞ are allowable values.)
and regularity conditions are satisfied.
• Theorem 10.4.2: If X1 , . . . , Xn is a random sample from a member of the REC(q1 , . . . , qk ),
then the statistics S1 =
n
X
i=1
t1 (Xi ),
S2 =
n
X
i=1
t2 (Xi ), . . . , Sk =
set of complete sufficient statistics for θ1 , . . . , θk .
n
X
i=1
tk (Xi ) are a minimal