Download homework 5.

Dr. Özlem İLK
Fall 2011-2012
IAM 530
HOMEWORK 5
Due 8 December 2011, Thursday, 9:40
You should work on these questions on your own. Please feel free to get help from me or from
Asena, but not from anyone else.
1. Electronic components of a certain type have a length of life Y with pdf
f ( y) 
1  y / 100
for y>0
e
100
Length of life is measured in hours.
a) Which specific distribution does this p.d.f correspond to? Please also state the
parameter(s) clearly.
b) Suppose that two such components (Y1 and Y2) operate independently and in parallel
(so, the system does not fail until both components fail) . Find the density function for
X, the length of life of system.
c) Suppose that two such components (Y1 and Y2) operate independently and in series
(so, the system fails when either component fails). Find the density function for X, the
length of life of system.
Hint: You need to consider order statistics for parts b) and c).
2. Suppose that X1 , X 2 ,... are i.i.d. continuous random variables having pdf
f(x)=x -2 for x>1.
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a) Find the CDF for a random variable having pdf f(x).
b) Let X(n) be the maximum among the first n random variables. Find the CDF of X(n).
Find the limiting distribution of X(n), if it exists.
c) Find the limiting distribution of X(n)/n, if it exists.
3. The annual number of earthquake registering 2.5 or higher on the Richter scale and
having an epicenter within 40 miles of downtown of city X follows a Poisson distribution
with λ=6.5.
a) Calculate the exact probability that nine or more such earthquakes will strike next
year.
b) Calculate an approximation to the probability mentioned in part a) based on central
limit theorem.
4. Suppose that X1 , X 2 ,... are i.i.d. random variables from Exp(1). Find the limiting
distribution of Yn=max( X1 , X 2 ,... )-ln(n), if it exists.
5. Let X1 , X 2 ,..., X n be a r.s. of size n from a distribution with pdf
f ( x;  ) 
1  | x  |
e
,    x  ,    
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a) Find MLE of θ .
b) Find MME of θ .
c) Is the estimator in part b) consistent?
Hint: You can use the following fact without proving: The median minimizes the average of
absolute deviations.
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