The number of calls coming per minute into a hotels reservation center is Poisso

Question

The number of calls coming per minute into a hotels reservation center is Poisson random variable with mean 3. (a) Find the probability that no calls come in a given 1 minute period. (b) Assume that the number of calls arriving in two different minutes are independent. Find the probability that at least two calls will arrive in a given two minute period. Let X equal the number of typos on a printed page with a mean of 3 typos per page. What is the probability that a randomly selected page has at least one typo on it?

Explanation / Answer

a.

Let X denote the number of calls coming in that given 1 minute period.

X ~ Poisson(3)
P(X = 0) =e-3*30/0!

=e-3

b.

Let X1 and X2 be the number of calls coming in the rst and second minutes rspectively.
We want

P(X1 + X2 >= 2).= 1 - P(X1 + X2 < 2)

=1 - [P(X1 = 0;X2 = 0) + P(X1 = 1;X2 = 0) + P(X1 = 0;X2 = 1)]
= 1 - [P(X1 = 0) * P(X2 = 0) + P(X1 = 1) * P(X2 = 0) +P(X1 = 0) *P(X2 = 1)]

=1-e-3e-3-e-3e-331/1!-e-3*31/1!

=1-7e-6

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