Question

2. During the summer months of June, July, and August, an average of 5 marriages per month take place in a small city. Assuming that these marriages occur randomly and independently of one another, and assuming a Poisson distribution, find the following: (a) The probability that 4 marriages will occur in June. (b) The probability that between 14 and 16 marriages (inclusive) will occur over the 3 months of June, July, and August (c) The probability that at at least 1 marriage will happen in a week of June (assume a month has 30 days)

Answer 1

Question 2

Here the number of marriages in these three months occuring is 5 marriages per month.

(a) If x is the number of marriages in June

so x ~ POISSON (5)

Pr(x = 4) = e^-5 5⁴/4! = 0.1755

(b) Here in cumulative three month expected number of marriages in three month = 5 *3 = 15

so if y is the number of marriages in three months

y ~ POISSON(15)

Pr(14 < = y < = 16) = Pr(y = 14) + Pr(y = 15) + Pr(y = 16)

= e^-15 15¹⁴/14! + e^-15 15¹⁵/15! + e^-15 15¹⁶/16!

= 0.1024 + 0.1024 + 0.0960 = 0.3009

(c) Here there are 4 weeks in a month of June

so expected number of marriges to occur in one week = 5/4 = 1.25 marriges per week

Here let say z is the number of marriages per week

Pr(at least 1 marriage per week) = Pr(z > = 1) = 1 - Pr(z = 0)

= 1 - e^-1.25 = 1 -0.2865 = 0.7135