Question

An internet service provider uses 50 modems to serve the needs of 1000 customers. It is estimated that at a given time. each customer will need a connection with probability 0.01, independent of the other customers.
What is the probability mass function of the number of modems in use at the given time?

Answer 1

Answer:

The probability mass function of the number of modems in use at the given time is:

$P (X=x)={1000\choose x}(0.01)^{x}(1-0.01)^{1000-x};\ x=0, 1, 2, ...49$

Step-by-step explanation:

Let the random variable X = number of modems in use.

The internet provider uses 50 modems.

The number of customers served by the internet user is, n = 1000.

The probability that a customer will require an internet connection is, p = 0.01.

It is provided that the customers are independent of each other.

The random variable X satisfies all the properties of a Binomial distribution with parameters n = 1000 and p = 0.01.

The probability mass function of a Binomial distribution is:

$P (X=x)={n\choose x}(p)^{x}(1-p)^{n-x};\ x=0, 1, 2, ...49$

Then the probability mass function of the number of modems in use at the given time is:

$P (X=x)={1000\choose x}(0.01)^{x}(1-0.01)^{1000-x};\ x=0, 1, 2, ...49$