Question

The local police department must write, on average, 5 tickets a day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 6.4 tickets per day.

Find the probability that less than 6 tickets are written on a randomly selected day from this population. __________

Find the probability that exactly 6 tickets are written on a randomly selected day from this population. ___________

Answer 1

Using the Poisson distribution, the probabilities are given as follows:

• Less than 6: 0.3837 = 38.37%.

• Exactly 6: 0.1586 = 15.86%.

What is the Poisson distribution?

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by:

$P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}$

The parameters are:

• x is the number of successes

• e = 2.71828 is the Euler number

• $\mu$ is the mean in the given interval.

For this problem, the mean is given by:

$\mu = 6.4$

The probability that exactly 6 tickets are written on a randomly selected day from this population is P(X = 6), hence:

$P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}$

[tex]P(X = 6) = \frac{e^{-6.4}(6.4)^{6}}{(6)!} = 0.1586/tex]

For less than 6, the probability is given by:

P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)

Using the same formula to find each value and adding them, we have that:

P(X < 6) = 0.3837.

More can be learned about the Poisson distribution at https://brainly.com/question/13971530

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