Due date: Friday 10/24, 9:00am
Functions on two random variables
01
03
PDF of a Min
Let
and be independent copies of a random variable. Let
. Find the PDF of . Link to originalSolution
12
(1) Write PDFs:
PDFs of
and : Joint PDF using independence:
(2) CDF of Min:
Write
. Then:
(3) Evaluate with formula for
: For
: Therefore:
Link to original
02
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
03
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
04
05
PDF of sum of uniforms
Let
and be independent copies of a random variable. Let . Find the PDF of
. Link to originalSolution
13
(1) Write PDFs:
PDFs of
and : Joint PDF using independence:
Method 1: CDF first
(2) CDF of sum:
Write
. Then: Plug in
. For
: For
:
(3) PDF from CDF:
Method 2: Convolution formula
(2) Convolution:
For
: For
: Link to original
Covariance and correlation
05
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
Review
06
01
Function on one variable
Suppose the PDF of
is given by: Find the CDF and PDF of
. Link to originalSolution
01
(1) Start by finding the CDF of
: Let us compute
: Thus, we have that:
(2) Finally, differentiate to find the density
: Link to original