HW#8, Due 05/04
AMS 311 Probability Theory (Spring 2011)
Read: Ross, Chapter 7, Section 7.2, 7.4, 7.7.
Part 1 (10 points):
PRINT
your name and StonyBrook ID (
Last, First, #ID
) on the top on your work
sheet. Staple your HW if more than 1 pages.
Part 2: Problems (90 points)
Write down your work step by step to get full score.
You do not need to evaluate
arithmetic expressions or integrals, if they are fully specified.
For example,
you may leave
in this form.
(1) (24 points)
(a) If E(3X) = var(X/2) and var(2X) = 3, find (i) E[(2 + X)
2
] and (ii) var(4 + 3X).
(b) Suppose that X and W are independent and that var(W) = 7, E((X − W)(X + W)) = 100,
E(3W) = 30, E(X
3
) = 60, and E(X +W) = 12. Compute (i) var(X) and (ii) cov(X
3
,W
2
)..
(2). (13 points) Let X have probability density function
Compute the moment generating function, M
X
(t) of X.
(3). (20 points) Consider a random variable X whose cumulative distribution function is given by
We are also told that P(X > 3.3) = 0.25.
(a) Find q, and compute the moment generating function, MX(t) of X.
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 Spring '08
 Tucker,A
 Normal Distribution, Probability theory, probability density function, moment generating function, lowest homework grade

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