Due date: Friday 10/10, 9:00am
Joint distributions
01
06
Stop lights
Let
be the number of stop lights and the number of red lights at which you must wait on your drive to grounds each day. The joint PMF of and is given below.
0 1 2 0 0.05 0.07 0.08 1 0.08 0.10 0.12 2 0.12 0.33 0.05 Find the marginal PMF of
and compute . Link to originalSolution
09
0 1 2 .25 .5 .25 Link to original
02
02
PMF 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 . Link to originalSolution
02
(a)
We know that
. Use this to find .
(b)
The marginal PMF of
is given by adding up the rows corresponding to each possible value.
(c)
(1) Define the possible values of
. Since
and , we have that .
(2) Define PMF of
. Go through each possible value of
and see when it occurs.
(3) Substitute values in for each probability.
(d)
(1) Find
(2) Find
Link to original
03
03
Marginals from joint PMF
Suppose the discrete joint PMF of
and is given by:
Compute the marginal PMFs
and . Link to originalSolution
03
(1) Compute
by summing over .
(2) Compute
by summing over . Link to original
04
05
Marginals and probability from joint PDF
Suppose
and have joint PDF given by: (a) Find the marginal PDFs for
and . (b) Find
. Link to originalSolution
05
(a)
(1) Find the marginal PDF for
by integrating the joint PDF with respect to .
(2) Find the marginal PDF for
by integrating the joint PDF with respect to .
(b)
Note that the region of interest is the one above the line
. Integrate the PDF over this region. Link to original
05
07
Grad student mentors
A university lab has an incoming group of researchers who need mentors. There are 4 graduate students, 2 undergraduate students, and the lab director who could serve as mentors. Suppose that exactly 3 of these 7 people will be selected as mentors.
(a) How many different groups of 3 people could be chosen to be the three mentors?
(b) Suppose exactly 2 graduate students and 1 undergraduate student are selected to be the three mentors. How many different groups of 3 people could be selected?
Let
be the total number of graduate students chosen to be mentors, and be the total number of undergraduate students chosen to be mentors. (c) Construct the joint PMF of
and : Link to originalSolution
11
(a)
(b)
(c)
Link to original
0 1 2 3 0 0 0 1 0 0 2 0 0
06
09
Air 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
. Link to originalSolution
13
0.09 0.11 0.05 0.25 0.17 0.23 0.08 0.48 0.06 0.15 0.06 0.27 0.32 0.49 0.19 (a)
(b)
Link to original
Independent random variables
07
04
Alice and Bob meeting at a cafe
Alice and Bob plan to meet at a cafe to do homework together. Alice arrives at the cafe at a random time (uniform) between 12:00pm and 1:00pm; Bob independently arrives at a random time (uniform) between 12:00pm and 2:00pm on the same day. Let
be the time, past 12:00pm, that Alice arrives (in hours) and be the time, past 12:00pm, that Bob arrives (in hours). So and represent 12:00pm. Find the joint PDF of
and . (Hint: and are independent.) Link to originalSolution
Review
08
05
Normal 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?
Link to originalSolution
14
(a)
(1) Write desired probability in terms of
values.
(2) Use lookup table to compute probability.
(b)
(1) Write desired probability in terms of
values.
(2) Use lookup table to compute probability.
(c)
(1) Set up conditional probability expression.
(2) Write desired probability in terms of
values.
(3) Use lookup table to compute probability.
Link to original
Optional challenge problem
- Complete this problem instead of one of the above problems of your choice.
- You must indicate on your paper which of the above problems should be replaced by this one. Indicate at the chosen problem (can otherwise leave blank) to direct the grader to the last page where you work your choice of challenge problem.
09
03
Composite PDF from joint PDF
The joint density of random variables
and is given by: Compute the PDF of
. Link to originalSolution
08
(1) Define
and find the CDF of .
(2) Differentiate to find
. Link to original