Cabs pass your workplace according to a Poisson process with a mean of five cabs per hour.

(a) Determine the mean and standard deviation of the number of cabs per 10-hour day.

(b) Approximate the probability that more than 65 cabs pass within a 10-hour day.

(c) Approximate the probability that between 50 and 65 cabs pass in a 10-hour day.

(d) Determine the mean hourly rate so that the probability is approximately 0.95 that 100 or more cabs pass in a 10-hour data.

Answer

Step 1 of 4</p>

(a)

Cabs pass your workspace according to a Poisson process with a mean of five cabs per hour.

We are asked to find the mean and standard deviation of the number of cabs per

Let denote the number of cabs per . Then, has a Poisson distribution with

……(1)

If is a Poisson random variable over an interval of length with parameter , then

Using equation (1), we can write the mean and variance,

Hence standard deviation,

Hence the mean and standard deviation of the number of cabs per is respectively.

Step 2 of 4</p>

(b)

We are asked to approximate the probability that more than pass within a

We need to find the probability

If is a Poisson random variable with and

Is approximately a standard normal variable.

The same continuity correction of used for the binomial distribution can also be applied.

The approximation is good for

This probability can be expressed exactly as

The computational difficulty is clear. Hence the probability can be approximated after continuity correction of as

From the z table, the area to the left of is .

Hence the approximate probability that more than pass within a is