Functions on two random variables
01 Theory
Theory 1
PMF of any function of two variables
Suppose
and are discrete RVs. The PMF of
: CDF of continuous function of two variables
Suppose
and are continuous RVs, and is a continuous function. The CDF of
: One can then compute the PDF of
by differentiation: Link to original
02 Illustration
Example - PDF of a quotient
PDF of a quotient
Suppose the joint PDF of
and is given by: Find the PDF of
. Solution
(1) Find the CDF using logic:
Integrate over this region:
(2) Differentiate to find PDF:
Compute
: Link to original
Exercise - PMF of
from chart PMF of XY squared from chart
Suppose the joint PMF of
and is given by this chart:
0.2 0.2 0.35 0.1 0.05 0.1 Define
. (a) Find the PMF
. (b) Find the expectation
Link to original.
Example - Max and Min from joint PDF
Max and Min from joint PDF
Suppose the joint PDF of
and is given by: Find the PDFs:
(a)
(b)
Solution
(a)
(1) Compute CDF of
: Convert to event form:
Integrate PDF over the region, assuming
:
(2) Differentiate to find
:
:
(b)
(1) Compute CDF of
: Convert to event form:
Integrate PDF over the region:
Therefore:
(2) Differentiate to find
:
: Link to original