Question

The number of hits on a certain website follows a Poisson distribution with a mean rate of 4 per minute. (a.) What is the probability that 5 hits are received in a given minute? (b.) What is the probability that 9 hits are received in 1.5 minutes? (c.) What is the probability that fewer than 3 hits are received in a period of 30 seconds?

Answer 1

Answer:

a) 15.63% probability that 5 hits are received in a given minute.

b) 6.88% probability that 9 hits are received in 1.5 minutes.

c) 67.67% probability that fewer than 3 hits are received in a period of 30 seconds.

Step-by-step explanation:

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

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

In which

x is the number of sucesses

$e = 2.71828$ is the Euler number

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

(a.) What is the probability that 5 hits are received in a given minute?

Mean rate of 4 per minute, which means that $\mu = 4$

This is P(X = 5).

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

$P(X = 5) = \frac{e^{-4}*(4)^{5}}{(5)!} = 0.1563$

15.63% probability that 5 hits are received in a given minute.

(b.) What is the probability that 9 hits are received in 1.5 minutes?

Mean rate of 4 per minute, so for 1.5 minutes, $\mu = 4*1.5 = 6$

This is P(X = 9).

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

$P(X = 9) = \frac{e^{-6}*(6)^{9}}{(9)!} = 0.0688$

6.88% probability that 9 hits are received in 1.5 minutes.

(c.) What is the probability that fewer than 3 hits are received in a period of 30 seconds?

Mean rate of 4 per minute, so for 30 seconds = 0.5 minutes, $\mu = 4*0.5 = 2$

This is

$P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)$

In which

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

$P(X = 0) = \frac{e^{-2}*(2)^{0}}{(0)!} = 0.1353$

$P(X = 1) = \frac{e^{-2}*(2)^{1}}{(1)!} = 0.2707$

$P(X = 2) = \frac{e^{-2}*(2)^{2}}{(2)!} = 0.2707$

So

$P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.1353 + 0.2707 + 0.2707 = 0.6767$

67.67% probability that fewer than 3 hits are received in a period of 30 seconds.