Due date: Tuesday 10/28, 11:59pm
Functions on two random variables
01
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
02
04
Coffee and danish
Joe visits a coffee shop every morning and orders a triple espresso and a cherry danish. Let
represent the wait time for the espresso and the wait time for the danish (both in minutes). The joint PDF of and is: Suppose that Joe will not sit down until he has both his espresso and his danish. Compute the probability that he will have to wait at least 5 minutes before sitting down, that is, find
. Link to originalSolution
03
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
04
02
Covariance etc. from independent densities
Suppose
and are independent variables with the following densities: Compute:
(a)
(b) (c) (d) Link to originalSolution
05
(a)
Compute
(b)
Compute
.
- Note that
and .
(c)
Compute
.
- Since
and are known to be independent, .
(d)
Compute
Link to original
- Since
, .
05
03
Plumber completion time
A plumber is coming to fix the sink. He will arrive between 2:00 and 4:00 with uniform distribution in that range.
Sink fixes take an average of 45 minutes with completion times following an exponential distribution.
When do you expect the plumber to finish the job?
What is the variance for the finish time?
Link to originalSolution
09
(1) Define random variables to describe the problem.
- Let
to represent the arrival time of the plumber. - Let
represent the completion time of the sink fix.
(2) Compute
- This represents the expected time the plumber finishes the job.
- Thus, we expect the plumber to finish at
.
(3) Compute the variance of the finish time.
Link to original
06
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
05
Variance puzzle: indicators
Suppose
and are events satisfying: Let
count the number of these events that occurs. (So the possible values are .) Find
. Hint: Try setting
. Link to originalSolution
07
(1) Define indicator variables
and Let
denote whether occurs Let
denote whether occurs. Note that they are independent since
and are independent ( ). Let
.
(2) Compute
. Link to original
08
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.
Review
09
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
08
Derived random variable
Let
and suppose that . (a) Compute
. (b) Compute
. (c) Compute
. Link to originalSolution
17
(a) From the sketch, we observe that
will be nonnegative. Hence for . Since has a uniform distribution on , for , . We use this fact to find the CDF of . For , For
and so The complete CDF can be written as
(b) By taking the derivative, the PDF is
Thus,
has an exponential PDF. In fact, since most computer languages provide uniform [ 0,1] random numbers, the procedure outlined in this problem provides a way to generate exponential random variables from uniform random variables.
(c) Since
Link to originalis an exponential random variable with parameter .
Optional challenge problem
- Complete this problem instead of two 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 problems (can otherwise leave blank) to direct the grader to the last page where you work the challenge problem.
11
06
When Suppose
for two random variables and . Show that
, where . Find the formula for
. Hint: Study the derivation that
, and think about . (Note: A similar result and argument holds for
.) Link to originalSolution
08
(1) Recall the formula for
. Therefore, if
, then , and note that .
(2) Compute
. Thus,
.
(3) Isolate
in the above equation. Thus,
Link to original, where .