CS173: Discrete Mathematical Structures
Spring 2006
Homework #9
Due 03/30/06, 8am
Solutions, Grading Rubric
1. (3 Points) Express E[X2] in terms of E[X] and Var[X].
Solution:
Var [X]=∑Xs− E[X]2Ps=...=E[X2]− E2[X]
E[X2]=Var [X] E2[X]
2. (6 points, 3 for base case, 3 for the step.)
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).
Solution:
Base case:
P(Ε1 ∩Ε2 )=1−P(¬Ε1 ∪¬Ε2 )=1−[P(¬Ε1 )+P(¬Ε2 )−P(¬Ε1 ∩¬Ε2 )]>=
1−[P(¬Ε1 )+P(¬Ε2 )]=1−[1−P(Ε1 )+1−P(Ε2 )]=P(Ε1 )+P(Ε2 )−1
Induction step:
P(Ε1 ∩Ε2 ∩...∩ΕΝ )=P([Ε1 ∩Ε2 ∩..ΕΝ −1].∩ΕΝ )>=[base case]>=
P([Ε1 ∩Ε2 ∩..ΕΝ −1])+P (ΕΝ )−1>=[hypothesis]>=
[P(E1)+P(E2)+...+P(EN-1 )-(n-2)]+P (ΕΝ )−1=P(E1)+P(E2)+...+P(EN )-(n-1)
3. (3 points) 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.
Solution:
1
26
8
4. (27 points, 2 points each leaf, except 4.3.4 worth 3 points)
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)?
Solution: 0
2. What is Pr(X = 1 ∧ X = 2)?
Solution: 0
2. If Pr(X∈{1, 2, 3}) = 0.9 and Pr(X∈{3,4,5,6}) = 0.5,
Docsity.com
