Written HW9

MTH 201 - Assignment 9 - written component
Assignment 9 is due at the beginning of class on Wednesday April 29th, 2015.
1. Let X be an exponential random variable with parameter λ = 1. Exponential random variables are very successful in describing times until the next failure
of an item in a large population of items. For the lifetime of an individual item
however, the exponential is often a reasonable fit but sometimes some modifications
are required to better fit the data. You will explore one such modification in this
problem.
1
(a) For fixed a, b, c > 0, let Y = a + bX c where X is exponential with λ = 1. Find
an explicit formula for the cumulative distribution function FY (y). Show your work.
(b) Use your answer in (a) to find the density function fY (y) for Y . Show your
work. Your answer should be a Weibull distribution discussed in section 5.6 of the
book. Find formulas for the parameters ν, α, β in the Weibull distribution you get
in terms of a, b, c.
24xy
if 0 < x < 1, 0 < y < 1, and 0 < x + y < 1,
2. Let f (x, y) =
0
otherwise.
(a) Show that f (x, y) is a joint probability density function for some continuous
random variables X and Y .
(b) Find E[X].
(c) Are X and Y independent? Justify your answer.
x+y
if 0 < x < 1, 0 < y < 1
3. The joint density function of X and Y is f (x, y) =
0
otherwise.
(a) Find the density function of X.
(b) Find P (X + Y < 1).
(c) Are X and Y independent? Justify your answer.
4. Let X and Y be two independent random variables. Often one has to consider
a new random variable W = min(X, Y ) which gives the smaller of the values of X
and Y .
(a) Find the cumulative distribution function FW (a) of W in terms of FX and FY .
(Hint: Notice that min(X, Y ) > a if and only if both X > a and Y > a. Now relate
the probability of the above event to FW (a)).
(b) Now assume further that X and Y are continuous random variables. Find a
formula for the density fW of W in terms of FX , fX , FY and fY .
5. Suppose X and Y are independent normal random variables with parameters µ1 ,
σ12 and µ2 , σ22 , respectively. Prove that X + Y is a normal random variable, and find
its parameter values.
6. A deck of n cards numbered 1 through n is thoroughly shuffled so that all possible
n! orderings can be assumed to be equally likely. Suppose you are to guess how the
cards are ordered as follows: you make n guesses sequentially, where your ith guess
is a guess of which card is in position i. Let N denote the number of correct guesses.
(a) If, while you are guessing, you are not given any information about your earlier
guesses, show that, for any strategy, E[N ] = 1.
(b) Suppose that after each guess you are shown the card that was in the position in
question. What do you think is the best strategy? Show that, under this strategy,
E[N ] =
1
1
1
1
+
+
+ · · · + + 1,
n n−1 n−2
2
and conclude that E[N ] ≈ log(n).
(c) Suppose that you are told after each guess whether you are right or wrong. In
this case, it can be shown that the strategy which maximizes E[N ] is one that keeps
on guessing the same card until you are told you are correct and then changes to a
new card. For this strategy, show that
E[N ] =
1
1
1
1
+
+
+ · · · + + 1,
n! (n − 1)! (n − 2)!
2!
and conclude that E[N ] ≈ e − 1.
Hint: For all parts, express N as the sum of indicator (that is, Bernoulli) random
variables.
7. An insurance company has 10,000 automobile policyholders. The expected yearly
claim per policyholder is $240, with a standard deviation of $800. Approximate the
probability that the total yearly claim exceeds $2.7 million. Explain any assumptions
you make about the underlying distributions.
8. Explain why a gamma random variable with parameters (α, λ) has an approximately normal distribution when α is large.
Hint: Consider α as a large integer, and recall how to interpret a gamma distributed
random variable in terms of waiting times for a Poisson distributed event. Then
invoke the central limit theorem.