The number of automobile accidents in Los Angeles per week,...

The number of automobile accidents in Los Angeles per week, $N$, has a Poisson distribution with mean 100 . The number of persons injured in each such accident has a binomial distribution with parameters $p=0.2$ and $N$. (a) What is the distribution of $Y$, the total number of persons injured in automobile accidents per week? (b) What is the mean and variance of $Y$ ?

The number of automobile accidents in Los Angeles per week, $N$, has a Poisson distribution with mean 100 . The number of persons injured in each such accident has a binomial distribution with parameters $p=0.2$ and $N$.
(a) What is the distribution of $Y$, the total number of persons injured in automobile accidents per week?
(b) What is the mean and variance of $Y$ ?

Key Concepts

Thinning Property of the Poisson Process

The thinning property states that if each event in a Poisson process is independently ‘accepted’ with a fixed probability, then the accepted events themselves form a Poisson process with a reduced rate. In the context of the problem, if every accident leads to an injury with a fixed probability, the overall count of injuries may itself follow a Poisson distribution with a mean equal to the original rate multiplied by that probability.

Compound Distributions

A compound distribution arises when one random variable is defined as the sum of a random number of independent and identically distributed (i.i.d.) random variables. In the context of the problem, the total number of injuries is a sum where the number of terms (accidents) is random (following a Poisson distribution) and each term (number of injuries per accident) follows a Binomial distribution.

Laws of Total Expectation and Variance

When dealing with hierarchical or compound distributions, the overall mean and variance can be computed using the laws of total expectation and total variance. These laws allow one to first condition on one random variable (for example, the number of accidents) and then average over its distribution, which is essential for finding the overall mean and variance of the number of injuries.

Poisson Distribution

A Poisson distribution models the number of times an event occurs 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. Its key properties include that its mean is equal to its variance, and it is often used to model count data such as the number of accidents in a given period.

Binomial Distribution

The Binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by parameters n (the number of trials) and p (the probability of success in each trial), and its mean is n·p with variance n·p·(1–p). This distribution is used here to model the number of injuries per accident.

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