The number of customers arriving per hour at a certain...

The number of customers arriving per hour at a certain automobile service facility is assumed to follow a Poisson distribution with mean $\lambda=7$. (a) Compute the probability that more than 10 customers will arrive in a 2 -hour period. (b) What is the mean number of arrivals during a 2-hour period?

The number of customers arriving per hour at a certain automobile service facility is assumed to follow a Poisson distribution with mean $\lambda=7$.
(a) Compute the probability that more than 10 customers will arrive in a 2 -hour period.
(b) What is the mean number of arrivals during a 2-hour period?

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Key Concepts

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, assuming that these events occur with a constant mean rate and independently of each other. This model is often used in scenarios where events happen sporadically over a continuous period, such as customer arrivals.

Time Scaling in Poisson Processes

When a Poisson process is observed over a time interval that is a multiple of the base period, the mean number of events scales linearly with time. For instance, if the average rate is known per hour, then over a two-hour period the mean number of events is doubled. This principle allows the adjustment of the distribution’s parameter to match different time intervals.

Complement Rule in Probability

The complement rule is a fundamental concept used in probability theory which states that the probability of an event occurring is one minus the probability of it not occurring. In problems where it is easier to calculate the probability of the complement (for example, calculating the probability of more than a certain number of events by subtracting the cumulative probability of that number or fewer events from one), this rule can simplify the solution process.

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