01
Finish a PMF table - Strange families
Suppose that 15 percent of the families in a strange community have no children, 20 percent have 1 child, 35 percent have 2 children, and 30 percent have 3 children. Assume the odds of a child being a boy or a girl are equal.
If a family is chosen at random from this community, then
, the number of boys, and , the number of girls, in this family will have the joint PMF partially shown in this table:
0 1 2 3 0 0.15 0.10 ? 0.0375 0.3750 1 0.10 0.175 ? 0.00 0.3875 2 ? ? 0.00 0.00 0.2000 3 0.0375 0.00 0.00 0.00 0.0375 0.3750 0.3875 0.2000 0.0375 [not used] (a) Complete the table by finding the missing entries.
(b) What is the probability that “
or is 1”?
Solution
11
(a)
Fill the cells using the respective column sum or row sum.
We have , so
: We have , so
: We have , so
: We have , so
(b)
(1) Add up the probabilities in which either
or .
(2) Alternatively, you could use the inclusion-exclusion principle using the marginal sums.
Link to original
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 .
Solution
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
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03
Marginals from joint PMF
Suppose the discrete joint PMF of
and is given by:
Compute the marginal PMFs
and .
Solution
03
(1) Compute
by summing over .
(2) Compute
by summing over . Link to original
04
Joint CDF on box events: All four corners
Consider the following formula:
Prove this formula. Hint: Do these steps along the way:
- Draw these events in the
-plane:
- Draw the event
. Write the probability of this event in terms of .
Solution
15
(1) Write the event
in terms of .
(2) Write the terms
, , and in terms of .
(3) Simplify expression for
. Link to original
05
Marginals and probability from joint PDF
Suppose
and have joint PDF given by: (a) Find the marginal PDFs for
and . (b) Find
.
Solution
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
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 .
Solution
09
0 1 2 .25 .5 .25 Link to original
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 :
Solution
11
(a)
(b)
(c)
Link to original
0 1 2 3 0 0 0 1 0 0 2 0 0
08
Sales rep making calls
A sales representative is responsible for selling a particular item. On a given day, he has time to make a sales pitch to up to 3 customers. His goal is to sell the item to 2 customers; if he is successful with the first two, he will not try to sell to the
customer. The probability of any one customer purchasing the item is 0.6, independent of the others. Let
be the number of customers to which he tries to sell the item and be the number of customers that purchase the item. Construct the joint PMF of and .
Solution
12
Link to original
2 3
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
.
Solution
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)
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10
Soft-drink machine
A soft-drink machine has a random amount
in supply at the beginning of a given day and dispenses a random amount during the day (with measurements in gallons). It is not resupplied during the day, and therefore . It has been observed that and have a joint density given by: (a) Find the probability that the amount of soft-drink dispensed on a given day is greater than 1.5 gallons. (First compute the marginal PDF of
.) (b) Find the probability that the amount of soda remaining in the machine at the end of the day is greater than
gallon. That is, find . (c) Find the CDF of
, the amount of soda remaining in the machine at the end of the day.
Solution
07
(a)
Marginal PDF of
is: Then:
(b)
(c)
Link to original
11
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. Recall the joint PDF of
and from the previous HW. (a) Consider
, the time that will pass (beyond 12:00pm) before Alice and Bob will begin working together. Find the CDF of W. (b) Find the probability Alice will have to wait more than 75 minutes for Bob.
Solution
09
(a)
Case 1:
:
Case 2:
:
(b)
Link to original
12
Joint PMF with
-dependence Suppose
and have the following joint PMF: (a) Find
. (b)
. Find .
Solution
16
(b)
Link to original