University of Illinois Spring 2008
ECE 313: Final Exam
Friday May 2, 2008
1. [48 points, 4 per answer]
In order to discourage guessing, 4 points will be deducted for each incorrect answer
(no penalty or gain for blank answers). A net negative score on this problem will
reduce your total exam score.
You do not need to show any work to justify your answers for this problem.
(a) Aand Bare two events such that 0 < P (A)<1 and 0 < P(B)<1.
Mark TRUE or FALSE for each question below.
TRUE FALSE
2 2 P(A∪B)≥max{P(A), P (B)}
2 2 P(A∩B)≥min{P(A), P (B)}
2 2 P(A|B) + P(A|Bc)=1.
2 2 P(A|B)P(B) + P(Ac|Bc)P(Bc)=1−P(A⊕B).
2 2 P(A|B)P(B) + P(Ac|B)P(B) = P(A).
2 2 If P(A) = P(B), then P(A|B) = P(B|A).
2 2 If P(A|B) = P(B|A), then P(A) = P(B).
2 2 If P(A|B) = P(A), then P(Bc|A)=1−P(B).
(b) Xand Yare random variables such that var(X) = var(Y) = σ2<∞.
Suppose that var(2X+ 3Y+ 4) = var(3X − 2Y+ 1).
Mark TRUE or FALSE for each of the following statements.
TRUE FALSE
2 2 Xand Yare uncorrelated random variables.
2 2 Xand Yare independent random variables.
2 2 var(2X+ 3Y+ 4) = var(2X − 3Y+ 1).
2 2 cov(X+Y,X − Y) = 0.
2. [42 points] At the Democratic National Convention (DNC), 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 DNC decides to
have a series of debates between the candidiates to decide the Democratic nominee.
Hillary wins a debate (event H) with probability p, and Barack wins a debate (event
B) with probability q= 1 −p.There are no draws. The debates continue until one
of the candidates wins two debates in a row and is declared the Democratic nominee.
Successive debates can be regarded as independent trials of an experiment, and X
denotes the total number of debates.
Express the answers to the following questions in terms of pand q, that is, do not
write (say) pq as p(1 −p) or multiply it out as p−p2. On the other hand, feel free to
simplify p+q= 1.
