Math 540/640: Statistical Theory I
Test II
Instructor: Songfeng (Andy) Zheng
Note: Please write your work on a piece of paper clearly. Please show your work in detail.
Problem 1: The number of defective components produced by a certain process follows
a Poisson distribution with mean k. Each defective component has probability p0as being
repairable.
a. Find the probability that exactly Ndefective components are found.
b. Suppose we have exactly Ndefective components, let Ybe the number of repairable
components, find the probability distribution of Y.
c. Find the moment generating function of the distribution you obtained in part b.
d. What is the probability that we find exactly Ndefective components, and among them,
nare repairable?
Problem 2: A continuous random variable Xhas probability density function f(x) as
f(x) = (ce−βx,if x > 0
0,otherwise
where β > 0 is a known constant, and cis an unknown constant.
a. Please find c.
b. Please find the cumulative distribution function.
c. Using the result from b, please find P(X > a +b|X > a), where a > 0 and b > 0 are
known constants.
d. Find the moment generating function of X, please also specify the range of tin the
moment generating function.
e. Using the result in d, find E(X).
Problem 3: Let Nbe a discrete random variable with all the possible values {2,3,· · ·}.
Let function p(n) be
p(n) = Cα
nαlog n,if n= 2,3,· · ·
and p(n) = 0 otherwise; where Cα>0 is a constant depending on α. For what values of α,
p(n) is a probability distribution? For what values of α, the expected value E(N) exists?
Please prove your conjecture. [You don’t need to evaluate Cαor E(N).]
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