University of Illinois Fall 2006
ECE 413: Problem Set 9
Due: Wednesday November 1 at the beginning of class.
Reading: Ross, Chapters 4 and 5
Reminder: No class on Friday October 27
Cancelled class will be made up on Monday October 30, 7-8 pm, 169 EL
This Problem Set contains six problems
1. Two discrete random variables Xand Yrespectively taking on values {u1, u2, . . .}and
{v1, v2, . . .}are called (unconditionally) independent random variables if for all uiand vj,
P[{X =ui} ∩ {Y =vj}] = P[{X =ui}]P[{Y =vj}].
Equivalently, the conditional pmf of Ygiven that X=uiis the same for all ui. The random
variables are conditionally independent given an event Aif for all uiand vj,
P[{X =ui} ∩ {Y =vj}|A] = P[{X =ui}|A]P[{Y =vj}|A].
Note that Amight be specified in terms of another random variable.
With this as prologue, let us return to Problem 1 of Problem Set 8. Let Wdenote the
number of α-particles that are not detected by the Geiger counter. Note that given that
{X =n},W=n− Y has conditional pmf Binomial(n, 1−p) and its unconditional pmf is
Poisson(λ(1 −p)).
(a) Explain why P[{Y =k} ∩ {W =l}|{X =n}] = 0 except when n=k+lwhen this
probability is k+l
kpk(1 −p)l.
(b) Are Yand Wconditionally independent given the event {X =n}?
(c) Use the theorem of total probability in conjunction with your answer of part (a) to write
down the value of P[{Y =k} ∩ {W =l}].
(d) Are Yand Wunconditionally independent?
2. Which of the following functions F(u) are valid CDFs? For those that are valid CDFs,
compute the probability that the absolute value of the random variable exceeds 0.5.
(a) F(u) =
0u < 0,
u2,0≤u < 1,
1, u ≥1.
(b) F(u) =
0u < 1,
2u−u2,1≤u≤2,
1, u > 2.
(c) F(u) = 1
2exp(2u)u≤0,
1−1
4exp(−3u), u > 0,(d) F(u) = 1
2exp(2u)u < 0,
1−1
4exp(−3u), u ≥0,
3. The number of hours that a student spends on ECE 440 homework is a random variable X
with CDF
FX(u) =
0, u < 0,
(1 + u)/8,0≤u < 1,
1/2,1≤u < 2,
(4 + u)/8,2≤u < 4,
1, u ≥4.
Note that this is a mixed random variable: it takes on some values with nonzero probability
(like a discrete random variable) but also takes on all values in intervals of the real line (like
a continuous random variable).
