The suicide rate in a certain state is 1 suicide per 100000

The suicide rate in a certain state is 1 suicide per 100,000 inhabitants per month. (a) Find the probability that, in a city of 400,000 inhabitants within this state, there will be 8 or more suicides in a given month. (b) What is the probability that there will be at least 2 months during the year that will have 8 or more suicides? (c) Counting the present month as month num- ber 1, what is the probability that the first month to have 8 or more suicides will be month number i, i >=1?

Solution

Let p be the suicidal rate.

p = 0.00001

a) City population = 400000

np = 4 = Mean

X - no of suicides in this city is Poisson with parameter =4

Hence

Prob ( there will be 8 or more suicides in a given month)

=P(X>=8)

= 0.0511 (POisson prob )

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b) For 2 months average = 8

P(X>=8 when average =8)

= 0.5470

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c) p = Prob for 8 or more suicides = 0.0511 (as per part a)

As each month is independent, and there are two outcomes,

X no of suicides in a particular month is

P(more than 8 suicides in a month i)

= Pro(<8 suicides in i-1 months) 0.0511

= (1-0.0511)^i-1(0.0511)

= (0.9489)^i-1(0.0511)