227k views
4 votes
A business has six customer service telephone lines. Let x denote the number of lines in use at any given time. Suppose that the probability distribution of x is as follows. Write each of the probabilities for x = 0, 1, 2, 3, 4, 5, and 6.

1 Answer

4 votes

Final answer:

To answer the student's question, the Poisson distribution formula is used to calculate the probabilities for the number of telephone lines in use at a given time. The Poisson distribution is suitable here because calls occur at a constant rate and independently of each other, making it simple to compute the probability for any number of calls.

Step-by-step explanation:

To calculate the probability that a business with six customer service telephone lines will be using a certain number of lines (denoted by x) at any given time, we can use the Poisson distribution. The Poisson distribution applies here because calls come in at a constant average rate, and the calls are independent of each other. Given a Poisson distribution with a mean (λ) which equals the average number of calls, the formula for calculating the probability of x calls is:

P(X = x) = (λ^x * e^-λ) / x!

Using this formula, we can calculate the probabilities for x = 0, 1, 2, 3, 4, 5, and 6. For example, if the average number of calls (λ) is 5.5, then the probability of receiving exactly 5 calls is:

P(X = 5) = (5.5^5 * e^-5.5) / 5!

To find the probability of receiving at most six calls, we would calculate the cumulative probability up to and including six calls:

P(x ≤ 6) = Σ from x=0 to 6 P(X = x)

We can use a calculator or software that has a Poisson cumulative distribution function (poissoncdf) to make these calculations easier.

User Janluke
by
8.8k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories