01
Constants in PDF from expectation
Suppose
has PDF given by: Suppose
. Find the only possible values for and . Then find .
02
Variance: Direct integral formula
Suppose
has PDF given by: Find
directly using the integral formula.
03
PDF of derived variable for
and Suppose the PDF of an RV is given by:
(a) Find
using the integral formula. (b) Find
, the PDF of (by calculating the CDF first). (c) Find
using . (d) Find
using results of (a) and (c).
04
Random point in
, PDF of and A random point is chosen in the unit square
. Outcomes are points
in this square. Events are regions in the square. The probability of a region is the area of . Define the random variable
by . This is just the -coordinate of the random point. Then the random variable is given by . This is just the squared -coordinate of the random point. (a) Describe the PDF of
. (b) Describe the PDF of
.
05
CDF of derived variable
Suppose
is a continuous random variable. Let . Find the CDF of .
06
Octane revenue
The owner of a small gas station has his 1,500 gallon tank of 93-octane gas filled up once at the beginning of each week. The random variable
is the amount of 93-octane the station sells in one week (in thousands of gallons). The PDF of X is shown below. Assuming the station consistently charges $3.00 per gallon for 93-octane and pays $2,000 for the weekly fill-up, find the CDF of
, the profit the station makes from the 93-octane in a week.
07
Square root
Suppose that
. Let
. Find the PDF of .
08
Derived random variable
Let
and suppose that . (a) Compute
. (b) Compute
. (c) Compute
.