University of Illinois Spring 2007
ECE 413: Problem Set 11 Due 4/18/07 at beginning of class
Jointly distributed random variables
Assigned reading: Ross, Sections 6.1-6.5
Noncredit exercises: Ross, Chapter 6: problems 1-23, theoretical exercises 1-17, self-test 1-15.
1. The CDF for a uniform distribution within a triangle
Suppose the random point (X, Y ) is uniformly distributed over the triangular region shown.
v
3
2
u
(a) Find the joint pdf of Xand Y. Be sure to specify fX,Y (u, v) for all (u, v) in the two-dimensional
plane, and not just for points inside the triangle.
(b) In a sketch of the entire (uo,vo) plane, indicate regions where the joint CDF FX,Y (uo, vo) is
zero, where it is one, where it is a function of uoonly, and where it is a function of voonly.
(c) Complete the following equation for the joint CDF:
FXY (uo, vo) =
0 if ???
??? if 0 ≤uo≤3 and 3vo≤2uo
??? if ???
??? if ???
1 if ???
2. A joint pmf with geometric distribution marginals
Suppose a plant tests wafers one at a time, exposing each wafer to phase one testing, and exposing
each wafer passing phase one testing to phase two testing. Wafers failing phase one testing are
discarded. Let p0denote the probability a wafer does not pass phase one, let p1denote the
probability a wafer passes phase one but not phase two, and let p2denote the probability a wafer
passes both phases. Note that p0+p1+p2= 1.Let Xbe the number of wafers tested until one
passes both phases, and let Ydenote the number of the first Xwafers that pass phase one (so the
count Yincludes the first wafer to pass both phases).
(a) Explain, using little or no calculation, why Yhas a geometric distribution, and identify the
parameter of the distribution.
(b) Find the joint pmf of Xand Y.
(c) Using your answer to part (b) and summation of a series, find the marginal pmf, pY. This gives
a second derivation of the fact that Yhas a geometric distribution.
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