Exercise 1.51 (* Mathematica 7.0.0 *)

(b) Calculating  P[N(1)<5 | N (0.5) > 2]

Denominator:  P[N (0.5) > 2]=

1-CDF[PoissonDistribution[2.5],2]

Numerator:  P[N (0.5) > 2, N(1)<5]=

PDF[PoissonDistribution[2.5],3]×CDF[PoissonDistribution[2.5],1]+PDF[PoissonDistribution[2.5],4]×CDF[PoissonDistribution[2.5],0]

P[N(1)<5 | N (0.5) > 2]=

((%)/(%%))

(c) largest λ: P[N(1)≥8]≤0.1 or P[N(1)≤7]≥0.9

FindRoot[CDF[PoissonDistribution[λ],7]==0.9,{λ,0.001}]