01
Constants in PDF from expectation
Suppose
has PDF given by: Suppose
. Find the only possible values for and . Then find .
Solution
06
(1) Recall formula for expectation of a continuous random variable.
(2) Use formula to find an equation relating
and .
(3) Recall that integrating a PDF should yield
. Integrate the PDF and find a second equation relating and .
(4) Solve system of equations for
and . Isolating
in the second equation yields . Plugging this expression into the first equation yields
. Solving for
yields 2.4, and thus .
(5) Compute variance using the formula
Compute
. Link to original
02
Variance: Direct integral formula
Suppose
has PDF given by: Find
directly using the integral formula.
Solution
07
(1) Recall the integral formula for variance.
Use the fact that
(2) Compute
.
(3) Compute
Link to original
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).
Solution
08
(a)
Compute
.
(b)
(1) Find the CDF of
.
(2) Find the CDF of
. Since
is monotone increasing, .
(3) Find the PDF of
by differentiating.
(c)
Find
.
(d)
Compute
. Link to original
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 .
Solution
09
Case 1:
Case 2:
Therefore:
Link to original
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.
Solution
10
Case 1:
Case 2:
Therefore:
Link to original
07
Square root
Suppose that
. Let
. Find the PDF of .
Solution
08
(1) PDF of
:
(2) CDF of
:
(3) PDF of
: Link to original
08
Derived random variable
Let
and suppose that . (a) Compute
. (b) Compute
. (c) Compute
.
Solution
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 .