Math 213 Exam 2 9Solutions)
Prof. I.Kapovich April 4, 2008
Problem 1[20 points]
For each of the following statements indicate whether it is true or false.
You DO NOT have to explain your answers.
(1) For any random variables Xand Ywe have V(X+Y) = V X +V Y .
(2) For S={−5,3,1,5}, ordered as listed, the permutation 3,1,5,−5
is a derangement.
(3) Let Abe the set of all bit-strings of length 10. Let Rbe a relation
on Aconsisting of all pairs (x, y)∈A×Asuch that the strings x
and yagree in their first two bits.
Then Ris an equivalence relation with exactly two equivalence
classes.
(4) Consider a recurrence relation an= 3an−1+n.
Then for every c, α, β ∈R,an=c3n+α+βn is a solution of the
above recurrence relation.
Answers:
(1) False. (The statement would be true if we knew that Xand Yare
independent random variables).
(2) True.
(3) False. (Although Ris an equivalence relation, it has four distinct
equivalence classes, corresponding to 4 distinct binary strings of length two
that can occur as initial segments of length two of strings from A).
(4) False. (One can find a particular solution of this recurrence relation
of the form an=α+βn, but the values of αand βthat work are not
arbitrary. In this case these specific values are α=−3
4,β=−1
2and the
general solution of the original recurrence relation is an=c3n−3
4−1
2, where
c∈Ris arbitrary.).
Problem 2[20 points]
Let n≥1 be an integer. A fair die is rolled ntimes, independently. Let
Z= 3X+ 2Ywhere Xis the number of tosses that came 1 up and where
Yis the number of tosses that came 6 up.
Find EZ and V Z .
Give all the details of your work.
Solution.
For i= 1,. . . , n let
Zi=(3,if the i-th toss is 1,
2,if the i-th toss is 6,0,otherwise.
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