University of Illinois Spring 2008
ECE 313: Hour Exam I
1. [30 points] Let A,B, and Cdenote three events defined on a sample space Ω,
and suppose that P(A) = 0.5, P (B) = 0.4, P (C) = 0.3, P (Bc∩C) = 0.2, and
P(A∩Bc∩Cc)=0.1.
Find the following probabilities: P(B∩C), P(B∩Cc), P((A∪B∪C)c), P(Ac∩B),
and P(Bc∩Cc).
If any probability cannot be computed from the given data, check the corresponding
box and leave the answer area blank.
2. (a) [10 points] If Xis a Poisson random variable with mean 4, what is var(2 + 3X)?
(b) [10 points] Let Ybe a negative binomial random variable with parameters (n, p)
where nis known, but the value of pis unknown. It is observed that {Y =k}.
What is the maximum-likelihood estimate ˆpof the parameter p?
3. (a) [8 points] Dilbert has 3 coins in his pocket, 2 of which are fair coins while the
third is a biased coin with P(H) = p6=1
2. The probability that a coin chosen at
random from his pocket will land Tails is 7
12 . What is the value of p?
(b) [18 points] Let Aand Bdenote events defined on a sample space. Given that
P(A)=3/5, P (B)=2/5, and P(B|A)=1/3, find P(A|B), P (Ac∪Bc) and
P(Bc|Ac).
4. [24 points] At the Democratic National Convention, Hillary Clinton and Barack
Obama have equal numbers of delegates committed to them, and neither candidate
can win the nomination on a ballot. In desperation, the Convention decides that the
two candidates shall debate each other and the winner shall be the nominee of the
Democratic Party. On a debate, Clinton wins (event H) with probability P(H), and
Obama wins (event B) with probability P(B). A draw (event D) occurs (that is,
neither wins) with probability P(D) = 1 −P(H)−P(B)>0. In case of a draw,
another debate is held. Successive debates can be regarded as independent trials, and
continue until either event Hor Boccurs, and the Democratic nominee is chosen.
Express the answers to the following questions in terms of P(H) and P(B).
(a) [8 points] What is the probability that Hillary is the Democratic nominee?
(b) [8 points] Given that no more than 5 debates were held, what is the conditional
probability that Obama won the nomination?
(c) [8 points] What is the expected number of debates at the Democratic National
Convention?