Due date: Tuesday 3/10, 11:59pm
Joint distributions
01
02
Link to originalPMF calculations from a table
Suppose the joint PMF of
and has values given in this table:
0 1 2 3 1 0.10 0.15 0 0.05 2 0.20 0.05 0.05 0.20 3 0.05 0 0.05 (a) Find
. (b) Find the marginal PMF of
. (c) Find the PMF of the random variable
. (d) Find
and .
Solution
02
05
Link to originalMarginals and probability from joint PDF
Suppose
and have joint PDF given by: (a) Find the marginal PDFs for
and . (b) Find
.
Solution
03
09
Link to originalAir pollution
In a certain community, levels of air pollution may exceed federal standards for ozone or for particulate matter on some days. In a particular summer week, let X be the number of days on which the ozone standard is exceeded, and let Y be the number of days on which the particulate matter is exceeded.
The following table represents the joint PMF for X and Y.
0.09 0.11 0.05 0.17 0.23 0.08 0.06 0.15 0.06 (a) Find
. (b) Find
.
Solution
04
03
Link to originalComposite PDF from joint PDF
The joint density of random variables
and is given by: Compute the PDF of
.
Solution
Independent random variables
05
5
Link to originalOne car outlasts the other
Suppose that
are two independent exponential random variables. (a) Find the joint PDF
. (b) Find the probability:
Compare to W05-B Q02 “Vehicle Lifetimes.” Which method is easier? (For yourself, not this HW.)
Review
06
05
Link to originalNormal distribution - cars passing toll booth
The number of cars passing a toll booth on Wednesdays has a normal distribution
. (a) What is the probability that on a randomly chosen Wednesday, more than 1,400 cars pass the toll booth?
(b) What is the probability that between 1,000 and 1,400 cars pass the toll booth on a random Wednesday?
(c) Suppose it is learned that at least 1200 cars passed the toll booth last Wednesday. What is the probability that at least 1300 cars passed the toll booth that day?
Solution