Download here.

Dr. Özlem İLK
Fall 2011-2012
IAM 530
HOMEWORK 3
Due 3 November 2011, Thurday, 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. Consider a sequence of independent coin flips, each of which has probability p of being Heads.
Define a random variable X as length of the run (of either Heads or Tails) starting from the first
trial. For example, x=3 if either TTTH or HHHT are observed. Find the distribution of X, and
find E(X).
2. Economic conditions cause fluctuations in the prices of raw commodities as well as in finished
products. Let X denote the price paid for a barrel of crude oil by the initial carrier, and let Y
denote the price paid by the refinery purchasing the product from the carrier. Assume that the
joint density for (X,Y) is given by
f(x,y)=c
20< x < y < 40
a) Find the value of c that makes this a joint density for a two-dimensional random variable.
b) Find the probability that the carrier will pay at least $25 per barrel and the refinery will
pay at most $30 per barrel for the oil.
c) Find the probability that the price paid by the refinery exceeds that of the carrier by at
least $10 per barrel.
d) Find the marginal densities for X and Y.
e) Find the probability that the price paid by the carrier is at least $25.
f) Find the probability that the price paid by the refinery is at most $30.
g) Are X and Y independent? Explain.
h) From a physical standpoint, should Cov(X,Y) be positive or negative? Why?
i) Find E(X), E(Y), E(XY).
j) Find E(Y-X). Interpret this expectation in a practical sense.
1
3. In a shopping mall the waiting time for an elevator is found to be uniformly
distributed between 1 and 5 minutes.
a) Write down the probability density function.
b) What is the probability of waiting no more than 3 minutes?
c) What is the probability that the elevator arrives in the first 30 seconds?
d) What is the probability of a waiting time between 2 and 3 minutes?
e) What is the expected waiting time?
4. An appliance store receives a shipment of 30 microwave ovens, 5 of which are (unknown to
the manager) defective. The store manager selects 4 ovens at random, without replacement, and
tests to see if they are defective. Let X=number of defectives found. Calculate the pmf and cdf of
X.
5. Please solve this question at hand, i.e. do not use computer.
Consider the ages of 8 people given below:
22, 37, 40, 28, 32, 26, 31, 27
The heights of these people are given below:
138, 179, 180, 135, 162, 154, 152, 167
a) Calculate the mean, median, variance and standard deviation of age.
b) Draw a box plot of age. You need to show the axis of your plot clearly. What is the shape
of the distribution? Are there any outliers?
c) Calculate the covariance and the coefficient of correlation between age and height.
d) Briefly describe what you have learned from the statistics you calculated.
2