{(1 (ii)
1
1 , 2 ) : Y (2 ) = 2} =
x=0
{(1 , 2 ) : X (1 ) = x, Y (2 ) = 2} .
Problem 7. Suppose X and Y are two random variables, and let f (x, y ) be their joint probability density
given as
X=1
Y=1
Y=2
1/3
0
X=2
1/3 .
1/3
Compute the marginal density for X and Y .
Problem 8. Let X be a discrete random variable that takes the value 0 with probability 1/2 and -1
with probability 1/4 and 1 with probability 1/4. Show that the random variable Y = X 2 is the indicator
function for the event that a fair coin lands heads in 1 toss.
Problem 9. Let be the space of outcomes obtained when a fair 6 faced dice is thrown 2 times. Let X
be the difference between the face in the rst toss and the second toss. Let Y be the sum of the faces in
the two tosses. Compute
(i) f , the joint density function for (X, Y )
(ii) fX , the marginal density for X
(iii) fY , the marginal density for Y
(iv) The set of outcomes for which X > 0 and Y
4.
Problem 10. Let X be a random variable and let the cdf of X be given by
x<0
0
0.22 0 x < 1 1
0.23 x < 2
F (x) =
0.5
2 x<3
0.9 3 x < 4
1
x 4.
Compute the pdf for X . Is X a continuous or a discrete random variable?
Problem 11*. (Optional, and not for the faint-of-heart.) This problem is very, very hard, and will not be
graded; it explores a connection between combinatorics, analysis and probability. If you are interested in
studying probability beyond this course, you may attempt it; however, not attempting it will not hinder
your ability to follow the material in this course. Some of the material in the problem is a preview of the
kind of things that you might do in the rst few weeks of a graduate course in analysis.
Let = P (N) = {0, 1}N = {innite sequences consisting of 0 and 1} and let P () be the collection of
all subsets of . Show that the following sequence of assertions is true.
(a) Let A . Show that A can be thought of as the binary expansion of a real number in the interval
[0, 1], and conversely.
(b) Use the previous part to show that there is a 1-1, onto, function : [0, 1].
2