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The second exam for a statistics course (stat 542) focusing on joint probability distributions, conditional distributions, independence, and moment generating functions. The exam includes five problems, covering topics such as calculating conditional distributions, finding marginal distributions, determining existence of expected values, and manipulating moment generating functions.
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Stat 542 Exam 2
November 8, 2005 Prof. Vardeman
1 , e^ if^ 0 and^0
0 otherwise
y x f x y y^ x^ y
⎛ ⎞ ⎜ ⎟ ⎝ ⎠
a) What are the conditional distributions of X | Y = y and of Y | X = x? (You should NOT have to do
any calculus to identify these.)
X | Y = y : Y | X = x :
b) What is the marginal distribution of Y?
c) E X does not exist. Carefully argue this.
d) For t > 0 completely set up but do not evaluate a double (iterated) integral giving
e) Argue carefully that the random variables XY and Y are independent.
2
2
(this is a simple case of a so called "random effects model" of applied statistics).
c) Argue carefully that the 3 random variables Y 1 (^) + Y 2 (^) + Y 3 (^) + Y 4 , Y 1 (^) − Y 2 , and Y 3 (^) − Y 4 are independent.
a) Suppose that
1
2
X 1 M t be the MGF of X (^) 1. It can be written in terms of M (^) X. Do this.
2
1 exp^ 2
t M t
2
exp exp
2
t t M t t
2 2 exp exp 2 2
2
t t t t
H t t
= is a MGF. Carefully argue this.
2 2 exp exp 2 2
2
t t t t
K t t
= is NOT a MGF. Carefully argue this.
