01
Random point in a triangle
Consider a joint distribution that is uniform over the triangle with vertices
, and . Suppose a point is chosen at random according to this distribution. (a) Find the joint PDF
. (b) Find the marginal PDFs for
and . (c) Are
and independent?
Solution
13
(a)
Find the area of the triangle, and find a formula for the PDF.
The area of the triangle is
. Therefore the PDF is
(b)
(1) Integrate with respect to
to find the marginal PDF for .
(2) Integrate with respect to
to find the marginal PDF for .
(c)
(1) Compute the product
and compare to .
(2) Consider the case wherein
Therefore,
Link to originaland are not independent.
02
Factorizing the density
Consider two joint density functions for
and : (Assume the densities are zero outside the given domain.)
Supposing
is the joint density, are and independent? Why or why not? Supposing is the joint density, are and independent? Why or why not? Correction Update: the
coefficient in is incorrect, it should be a constant which is irrational. If you wish, you may instead substitute as the formula over the same domain for .
Solution
14
(1) Compute the marginal distribution of
by integrating with respect to .
(2) Compute the marginal distribution of
by integrating with respect to .
(3) Determine independence by multiplying the marginal pdfs.
Since the product of the marginal PDFs equals the joint PDF, we conclude that
and are independent.
(4) Compute the marginal distribution of
by integrating with respect to .
(5) Compute the marginal distribution of
by integrating with respect to .
(6) Determine independence by multiplying the marginal pdfs.
Therefore,
Link to originaland are not independent.
03
Composite PDF from joint PDF
The joint density of random variables
and is given by: Compute the PDF of
.
Solution
08
(1) Define
and find the CDF of .
(2) Differentiate to find
. Link to original
04
Alice and Bob meeting at a cafe
Alice and Bob plan to meet at a cafe to do homework together. Alice arrives at the cafe at a random time (uniform) between 12:00pm and 1:00pm; Bob independently arrives at a random time (uniform) between 12:00pm and 2:00pm on the same day. Let
be the time, past 12:00pm, that Alice arrives (in hours) and be the time, past 12:00pm, that Bob arrives (in hours). So and represent 12:00pm. Find the joint PDF of
and . (Hint: and are independent.)
Solution