The number of telephone calls that arrive at a phone exchange is often modeled as a Poisson random variable. Assume that on the average there are 10 calls per hour. (a) What is the probability that there are three or fewer calls in one hour

Answer: the probability that there are three or fewer calls in one hour is 0.011

Step-by-step explanation:

The formula for poisson distribution is expressed as

P (x = r) = (e^ - µ * µ^r) / r!

Where

µ represents the mean of the theoretical distribution.

r represents the number of successes of the event.

From the information given,

µ = 10

For the probability that there are three or fewer calls in one hour, it is expressed as

P (x ≤ 3) = P (x = 0) + P (x = 1) + P (x = 2) + P (x = 3)

Therefore,

P (x = 0) = (e^ - 10 * 10^0) / 0! = 0.000045

P (x = 1) = (e^ - 10 * 10^1) / 1! = 0.00045

P (x = 2) = (e^ - 10 * 10^2) / 2! = 0.0023

P (x = 3) = (e^ - 10 * 10^3) / 3! = 0.0077

Therefore,

P (x ≤ 3) = 0.000045 + 0.00045 + 0.0023 + 0.0077 = 0.011