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Theoretical Statistics
Department of Mathematics
Ph.D. Qualifying Examination
Time: 240 minutes
January 03, 2013
General Instructions:
• There are two parts in this examination. Part A (STA 584) has 6 questions and Part B
(STA 684) has 6 questions.
• Begin each question on a new sheet with the question number clearly labeled. Write on
one side only. When finished, please arrange all pages according to the question numbers
and then number the pages accordingly.
• You must show all your work correctly to earn full credits. Partial credits will be given
for partially correct solutions.
Part A [Answer all questions. Questions are not equally weighted]
Question #01
A test for a disease correctly diagnoses a diseased person as having the disease with
probability 0.95. The test incorrectly diagnoses someone without the disease as having the
disease with a probability of 0.05. If 1% of the people in a population have the disease, what
is the chance that a person from this population who tests positive for the disease actually has
the disease?
Question #02
Let X and Y be discrete random variables with joint probability function
 y
for x = 1,2,4; y = 2,4,8, and x ≤ y

f ( x, y ) =  24 x

elsewhere
0
(a) What is the probability that Y – X is greater than 5?
(b) What is the marginal distribution of X?
(c) What is the cumulative distribution function of X?
(d) Calculate the variance of Y given X = 4.
(e) Define U = Y / X and V = X. What is the joint probability distribution of U and V?
Question #03
A random variable X has a gamma distribution if and only if its probability density function
kxα −1e − x / β , for x > 0
is given by f ( x) = 
where α > 0 and β > 0.
elsewhere,
0,
(a) Find k.
(b) Calculate the moment generating function of X.
(c) Use your result of part (b) to find the mean and variance of X.
(d) Use your result of part (b) to find the moment generating function of an exponential random
variable with a mean of λ .
(e) Use your result of part (c) to find the mean and variance of a chi-square random variable with
n degrees of freedom.
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Question #04
The joint distribution of X and Y is given by
 xy, for 0 < x < 2 and 0 < y < 1
f ( x, y ) = 
0, elsewhere.
(a) What is the probability that X + Y is greater than 1?
(b) What is the probability that X is greater than 1?
(c) What is the conditional probability distribution of Y given X = x, where 0 < x < 2?
(d) Find the variance of Y given X = x, 0 < x < 2.
Question #05
The joint distribution of X and Y is given by
3
 x, for 0 < x < y < 2
f ( x, y ) =  4
0,
elsewhere.
(a) Find marginal distribution of Y.
(b) Find the covariance of X and Y.
Question #06
Define U = Y / X and V = X, where X and Y are independent exponential variables each with
a mean of 1.
(a) Find the joint distribution of U and V.
(b) Find the marginal distribution of U.
Part B [Answer all questions. Questions are not equally weighted]
Question #07
(a) Let a continuous random variable X have probability distribution for which the mean µ and
variance σ 2 are finite. Then for every k > 0, prove that P ( X − µ ≤ kσ ) ≥ 1 − k −2 .
(b) Let Yn denote the maximum of a random sample of size n from a distribution that has the
probability density function=
Z n n(Yn − θ ) . Does Z n
f ( x) e x −θ , x ≤ θ , zero elsewhere. Let =
converge in distribution to some random variable Z? If so, find the probability density
function of Z.
(c) For the following sequences of independent random variables, does the weak law of large
numbers hold?
(i) P ( X n =
±2n ) =
1/ 2
(ii)
P( Xn =
±2n ) =
2−2 n −1 , P ( X n ==
0 ) 1 − 2−2 n
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Question #08
(a) State without proof the Central Limit Theorem.
(b) Let X n denote the mean of a random sample of size n from a distribution that has the
probability mass function f ( x) =
θ x (1 − θ )1− x , x =
0, 1 .
(i) Find the moment generating function M(t) of X.
(ii) Find the moment generating function M (t , n) of Yn = n ( X n − θ ) / θ (1 − θ ) .
(iii) By taking the limit of the moment generating function M (t , n) in (ii), determine the
limiting distribution of Yn as n → ∞ . Identify the parameters of the limiting
distribution of Yn .
Question #09
(a) Let X be a random variable of the continuous type with probability density function f ( x) ,
which is positive provided 0 < x < a < ∞ , and is equal to zero elsewhere. Show that
E (=
X)
∫
a
0
[1 − F ( x)]dx , where F ( x) is the cumulative distribution function of X.
(b) Two numbers are selected at random from the interval (0, 1). If these values are uniformly
and independently distributed, by cutting the interval at these numbers compute the
probability that the three resulting line segments form a triangle.
(c) Let 0 < p < 1. A (100p)th percentile (quantile of order p) of the distribution of a continuous
random variable X is a value ξ p such that P( X ≤ ξ p ) =
p . Consider a random variable that
has the probability density function f ( x) = 5 x 4 for 0 < x < 1, and zero otherwise.
(i) Find ξ p in terms of the probability p.
(ii) Find the 80th percentile of the distribution.
(iii) A CMU faculty’s salary is at the 80th percentile. Explain the meaning of this percentile.
(iv) Using your result in (i) or otherwise, what percentile is x = 0.99?
Question #10
(a) What is a consistent estimator?
(b) Let X 1 , X 2 , , X n denote a random sample from uniform distribution with probability
1/ (3 − 2θ ), 2θ < x < 3
density function f ( x;θ ) = 
elsewhere.
0,
(i)
(ii)
(iii)
(iv)
(v)
Is θ > 0 always true? If so, explain and if not state the possible values of θ.
Find the moment estimator of θ .
Find the maximum likelihood estimator of θ .
Find an unbiased maximum likelihood estimator of θ .
Show that the moment estimator in (ii) is consistent.
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Question #11
(a) Define an exponential family of probability density functions and give an example to
illustrate your definition.
(b) Suppose a random sample of size n is taken from the generalized negative binomial
m  m + 2x  x
distribution with probability function
=
f ( x;θ )
− θ ) m + x , x 0,1, 2,3, ,

 θ (1=
m + 2x  x 
where 0 < θ < 0.5 and m > 0 are parameters. The population mean µ and variance σ 2 for the
distribution are
=
µ mθ (1 − 2θ ) −1 and σ 2 = mθ (1 − θ )(1 − 2θ ) −3 respectively.
(i)
Does the generalized negative binomial distribution belong to the exponential family?
Justify your answer.
(ii) If the parameter m is known, find the moment estimator of θ.
(iii) Find the moment estimators of θ and m. [Hint: Estimator of θ should have two roots.]
(iv) Determine the values of θ for which the population variance σ 2 is greater than the
population mean µ. Based on your answer, is σ 2 always greater than µ for the
distribution? Explain. Using this result or otherwise, determine the correct estimator of
θ from the two roots in (iii).
Question #12
(a) State the Neyman-Pearson theorem.
(b) Suppose a random sample of size n is taken from generalized Poisson distribution with the
x −1
probability function f ( x;θ ) =
(1/ x !) (1 + 0.2 x ) θ x e −θ (1+ 0.2 x ) , x =
0,1, 2,3, , and θ > 0 is a
parameter. Consider the simple null hypothesis H 0 : θ = 1.0 against the alternative hypothesis
H1 : θ = 0.5 .
(i) Use the Neyman-Pearson theorem to find the best critical region of size α.
(ii) Show that the best critical region can be written in term of the sample mean X .
(iii) Suppose n = 1, can one find a constant k for the best critical region such that α is less
than 0.2? Explain.
(c) Let Y1 < Y2 <  < Y5 be the order statistics of a random sample of size n = 5 from a
distribution with probability density function f ( x;θ ) = (5θ ) −1 , 0 < x < 5θ , zero elsewhere,
where θ > 0 . The hypothesis H 0 : θ = 1 is rejected and H1 : θ < 1 is accepted if the observed
Y5 ≤ k .
(i) Find the constant k so that the significance level α = 0.05.
(ii) Determine the power function of the test under the alternative hypothesis.
(iii) Can you use the function in (ii) to compute the power at θ = 1.5? Why or why not?