Joint distributions
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
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
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
12
Joint PMF with
-dependence Suppose
and have the following joint PMF: (a) Find
. (b)
. Find . Link to originalSolution
16
(b)
Link to original
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. Link to originalSolution
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.
Link to originalSolution
09
(a)
Case 1:
:
Case 2:
:
(b)
Link to original
Independent random variables
01
Random point in a triangle
Consider a joint distribution that is uniform over the triangle with vertices
, and . Suppose a point is chosen at random according to this distribution. (a) Find the joint PDF
. (b) Find the marginal PDFs for
and . (c) Are
and independent? Link to originalSolution
13
(a)
Find the area of the triangle, and find a formula for the PDF.
The area of the triangle is
. Therefore the PDF is
(b)
(1) Integrate with respect to
to find the marginal PDF for .
(2) Integrate with respect to
to find the marginal PDF for .
(c)
(1) Compute the product
and compare to .
(2) Consider the case wherein
Therefore,
Link to originaland are not independent.
Functions on two random variables
02
PDF of Min and Max
Suppose
and and these variables are independent. Find: (a) The PDF of
(b) The PDF of
Link to originalSolution
02
(a)
(1) First, we define the PDF’s and CDF’s of
and :
(2) Now:
since, by definition,
. By independence:
Thus, we have:
and thus,
(b)
(1) Similarly, for
, we have that . Thus, we have: by independence. Thus, we have that:
(2) Thus, the density function is given by:
Link to original
01
PDF of sum from joint PDF
Suppose the joint PDF of
and is given by: Find the PDF of
. Link to originalSolution
03
(1) Let
. We want to find , which we shall do using the convolution formula. Loosely, we have that for acceptable values of a ^7hebc8nd .
(2) First, consider the range of
: Since and , we have that . Thus, we need only concern ourselves with the case when .
(3) Now that we have a range for
, we must now find acceptable values of . Since both and , we have that . However, , by the condition for the JPDF given above. Thus, .
(4) Similarly,
and . Solving the second equation, we have that . Thus, . Since , , . Thus, we can restrict our condition to .
(5) Now that we have bounds, we can finally apply the convolution formula:
(6) We now take cases to deal with the upper bound: when
, , and so our upper bound is . If , and , so our upper bound is . Plugging these values in and evaluating, we have our density function: Link to original
03
PDF of sum of arbitrary uniforms
Suppose that:
and are independent Find the PDF of
in terms of the parameters . You may assume that . Link to originalSolution
04
(1) Convolution formula:
The range of
is .
(2) Divide into cases:
:
:
: (Note: there are
satisfying this inequality because we assume .)
:
.
(3) Write out piecewise function:
Link to original
06
Lights on
An electronic device is designed to switch house lights on and off at random times after it has been activated. Assume the device is designed in such a way that it will be switched on and off exactly once in a 1-hour period. Let
denote the time at which the lights are turned on and Y the time at which they are turned off. Assume the joint density for is given by: Let
, the time the lights remain on during the hour. (a) Find the range of
. (b) Compute a formula for the CDF of
, i.e. . (c) Find the probability the lights remain on for at least 40 minutes in some given hour.
Link to originalSolution
15
(a) Range of
is .
(b)
Alternate:
(c)
Link to original
Covariance and correlation
01
Covariance and correlation
The joint PMF of
and is given by the table:
0 1 2 3 1 2 3 0 Compute:
(a)
(b) (c) (d) Link to originalSolution
11
(a)
Note that the formula for
.
(b)
Compute
using the same formula.
(c)
(1) Recall the formula for
.
(2) Compute
and .
(3) Compute
.
(4) Compute
.
(d)
(1) Recall the formula for
.
(2) Compute
and .
(3) Compute
and .
(4) Compute
Link to original
04
Correlation between overlapping coin flip sequences
Suppose a coin is flipped 30 times.
Let
count the number of heads among the first 20 flips, and count the heads in the last 20. Find
. Hint: Partition the flips into three groups of 10. Create three variables, counting heads, and express
and using these. What is the variance of a binomial distribution? Link to originalSolution
06
(1) Define random variables for partitioning the 30 flips into groups of 10.
Let
be the number of heads in the first 10 flips. Let
be the number of heads in the middle 10 flips. Let
be the number of heads in the last 10 flips. Clearly,
and are independent. Note that
and .
(2) Compute
and
(3) Compute
(4) Compute
Since
and , . Thus,
(5) Compute
(6) Compute
. Link to original
07
Covariance etc. from joint density
Suppose
and are random variables with the following joint density: Compute:
(a)
(b) (c) (d) (e)
(f) (g) (h) Are and independent? (It is worth thinking through which of these can be computed in multiple ways.)
Link to originalSolution
10
(a)
Compute
by integrating the joint PDF.
(b)
Compute
.
(c)
Compute
(d)
(1) Compute
(2) Compute
.
(e)
(1) Compute
(2) Compute
.
(f)
Compute
(g)
Compute
.
(h)
Determine independence.
Link to original
- Since
, we can conclude that and are not independent.
Conditional distribution
TBD
Conditional expectation
TBD