University Problem Set #7 ECE 313
of Illinois Page 1 of 3 Summer 2003
Assigned: Thursday, July 24
Due: Thursday, July 31
Reading: Ross Chapters 6.1-6.4
Problems:
1. The probability of heads of a random coin is a random variable P uniform in the interval
(0,1).
(a) Find P{0.3 ≤ P ≤ 0.7}.
(b) The coin is tossed 10 times and heads shows 6 times. Find the posteriori probability that
P is between 0.3 and 0.7.
2. We place at random 100 points in the interval (0,100). Find the probability that in the
interval (0,0.1) there will be one and only one point
(a) Exactly
(b) Using the Poisson approximation
3. The discrete random variables X and Y have joint pmf pX,Y(u,v) given by
4 0 1/12 1/6 1/12
3 1/6 1/12 0 1/12
–1 1/12 1/6 1/12 0
v↑ u→ 0 1 3 5
(a) Find the marginal pmfs pX(u) and pY(v) of X and Y.
(b) Are the random variables X and Y independent ?
(c) Find P{X ≤ Y} and P{X + Y ≤ 8}.
(d) Find pX|Y(u|3), E[X|Y=3], and var(X|Y=3).
4. The number of α-particles emitted by a source during a unit time interval can be modeled
as a Poisson random variable X with parameter λ. The α-particles are detected by means
of a (imperfect) Geiger counter which detects a particle with probability p < 1. The
detections of the various particles can be considered to be independent events. Thus, if n
particles have been emitted, the Geiger counter reading can be modeled as a binomial
random variable Y with parameters (n,p). In short, pY|X(k|n), the conditional pmf of Y
given that X = n, is a binomial pmf: pY|X(k|n) =
n
k pk(1–p)n–k for 0 ≤ k ≤ n. (Note that
Y ≤ X always: the counter does not mistakenly count a particle when no particle is
present, i.e. there are no false alarms! but the counter does occasionally fail to detect a
particle).
(a) Sketch the u-v plane and the joint pmf of X and Y. Precision is not required in the sizes
of the blobs you draw, but be sure that you don’t put masses where they do not belong.
(b) What is the unconditional pmf of Y?
(c) What is the conditional pmf of X given Y = k?
5. The jointly continuous random variables X and Y have joint pdf given by
f
X,Y(u,v) =
2 exp –(u + v),0 < u < v < ∞,
0, elsewhere.
(a) Sketch the u-v plane and indicate on it the region over which fX,Y(u,v) is nonzero.
(b) Find the marginal pdfs of X and Y.
(c) Are the random variables X and Y independent ?
(d) Find P{Y > 3X}.
(e) For α > 0, find P{X + Y ≤ α}.
