CS173: Discrete Mathematical Structures
Spring 2006
Homework #9
Due 03/30/06, 8am
1) Express E[X2] in terms of E[X] and Var[X].
2) Let E1,E2,...,EN be N events from finite sample space. prove by induction that
P(Ε1 ∩Ε2 ∩...∩ΕΝ )>=P(E1)+P(E2)+...+P(EN )-(n-1).
3) A computer text generation program randomly chose a letter (out of the 26) 8 times(with
repetitions) with uniform probability and the word was “mananaga”. What was the probability
of generating this word? Give an expression with explanation, not a number.
4) We are given a random variable X whose domain is {1, 2, 3, 4, 5, 6} (each number represents
the event of throwing a die that lands with that number facing up).
1. If Pr(X = 1) = 1,
1. What is: Pr(X = 2)?
2. What is Pr(X = 1 ∧ X = 2)?
2. If Pr(X∈{1, 2, 3}) = 0.9 and Pr(X∈{3,4,5}) = 0.5,
1. What is the lower bound on Pr(X∈{1, 2, 4})?
2. What is the upper bound on Pr(X∈{1, 2, 4})?
3. If Pr(X = 1) = Pr(X = 2) = ... = Pr(X = 6), what is:
1. Pr(X = 2 ∧ X = 5)?
2. What is Pr(X =1) + Pr(X = 2) + Pr(X = 3)?
3. E[X]?
4. Var[X]?
5. What is the expected number of times needed to throw a die to achieve 6 at least once?
Prove your answer. Answer without proof will not be accepted.
4. Assume that we are given another random variable Y , and know that X, Y are have
identical, independent distributions, as defined in 4.3.
1. What is Pr(X = 1|Y = 1)?
2. What is Pr(X = 1, Y = 1)?
3. What is Pr(X = 4 ∨ X = 6 | X+Y=10)?
4. What is Pr(X = 4 ∨ Y = 6 | X+Y=10)?
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