The suicide rate in a certain state is 1 suicide per 100,000...

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 number $1,$ what is the probability that the first month to have 8 or more suicides will be month number $i, i \geq 1 ?$ What assumptions are you making?

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 number $1,$ what is the probability that the first month to have 8 or more suicides will be month number $i, i \geq 1 ?$ What assumptions are you making?

Key Concepts

Geometric Distribution

The Geometric distribution describes the number of independent trials needed for the first success to occur. It applies to scenarios where one is looking for the probability that a specific event will first occur on a particular trial, such as determining the first month in which an event count exceeds a threshold, as seen in part (c).

Poisson Distribution

The Poisson distribution is used to model the number of occurrences of an event in a fixed interval of time or space when these events occur with a known constant mean rate and independently of the time since the last event. It is especially appropriate for rare events and can be used to calculate probabilities for a given count in a specific period, as required in part (a).

Binomial Distribution

The Binomial distribution models the number of successes in a fixed number of independent trials with the same probability of success. In the context of the problem, it is used to determine the probability of achieving a certain number of months (out of a total number of months) where a particular event count exceeds a threshold, which is exemplified in part (b).

Assumptions of Independence and Stationarity

When applying these probability models, it is assumed that events occur independently over time and that the probability characteristics do not change from one interval to the next. This means that individual time periods (months, for instance) are treated as independent trials with a constant event rate, which justifies the use of Poisson, Binomial, and Geometric distributions in the analysis.

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