STAT/MATH 394B

Solutions, Problem Session 9

1. Possible outcomes are BB, BG, and GB, all equally likely, so the answer is 1/3.

2. P(X>x)=.

3.

The required probability is 0.52/(0.52+0.07)=0.88.

4. (a) No, gets larger than 1.

(b) No, decreases at x=1/4

(c) Yes, nondecreasing, goes to 0 at negative infinity, to 1 at plus infinity.

5. (a) P(e₁)=rp; P(e₂)=(1-r)p²; P(e₃)=(1-r)p(1-p); P(e₄)=(1-r)(1-p)p; P(e₅)=r(1-p); P(e₆)=(1-r)(1-p)².

(b) A={e₁,e₂,e₃}; B={e₁,e₅} so P(A|B) = P{e₁)/(P(e₁)+P(e₅))=rp/(rp+r(1-p))=rp/r=p, while P(A)=rp+(1-r)p(p+1-p)=rp+(1-r)p=p. Hence P(A|B)=P(A), so A and B are independent.