01
PDF of sum from joint PDF
Suppose the joint PDF of
and is given by: Find the PDF of
.
Solution
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
02
Poisson plus Bernoulli Suppose that:
and are independent Find a formula for the PMF of
. Apply your formula with
and to find .
Solution
04
(1) We have that
(2) Let
. Then: Now, since
only if , we have that . Thus,
(3) Plugging in
Link to original, , and into our PDF above, we have that .
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 .
Solution
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
04
Sums of normals (a) Suppose
are independent variables. Find the values of and for which , or prove that none exist. (b) Suppose
, in part (a). Find . (c) Suppose
and . Find .
Solution
06
(a)
(1) Suppose
and are independent (since they have the same distribution, they are called independent identically distributed, or IID, random variables). Now suppose that . Notice that if , then and .
(2) For the first condition, we have:
and thus the first condition holds for any
.
(3) For the second condition, by independence, we must have
. Then we must have . Since , we must have that as the only solution.
(4) Thus,
and , which signals that are constants, are the only values that satisfy the given condition.
(b)
(1) Define
. The mean is
. The variance is
Thus,
.
(2) Note that
. Standardize and use the lookup table.
(c)
(1) Let
.
(2) Note that
. Standardize and use the lookup table. Link to original
05
PDF of sum of uniforms
Let
and be independent copies of a random variable. Let . Find the PDF of
.
Solution
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
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.
Solution
15
(a) Range of
is .
(b)
Alternate:
(c)
Link to original
