01
PDF of sum from joint PDF
Suppose the joint PDF of
and is given by: Find the PDF of
.
02
Poisson plus Bernoulli Suppose that:
and are independent Find a formula for the PMF of
. Apply your formula with
and to find .
Solution
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 .
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
05
PDF of sum of uniforms
Let
and be independent copies of a random variable. Let . Find the PDF of
.
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.