Question

Data indicates that the number of traffic accidents on a rainy day is Poisson with mean 19, while on a dry day, it is Poisson with mean 13. Let X denote the number of traffic accidents tomorrow. Suppose that the chance it rains tomorrow is 0.6. Find(a)P(rain tomorrow|X= 15) using Bayes rule. (b)E(X).

Answer 1

a)P(rain tomorrow|X= 15) using Bayes rule as consider by:

= P(X=15)

= P(X=15 and rainy day) + P(X=15 and dry day)

= (e^-19*19^15/15! ) *0.6 + ( e^-13*13^15/15!)*0.4
=   0.0650*0.6 + 0.0885*0.4

= 0.07441673

P (rain tomorrow|X= 15) = 0.0650*0.6/0.07441673

                                    =    0.5244

Therefore,

P(x =15) = 0.5244

b) E(X) as consider by:

E(x) = 19*0.6+13*0.4

        = 16.6

Therefore,

E(x) = 16.6