Need help on a and b The number \( x \) of people entering the intensive care unit at a particular hospital on any one day has a Poisson probability distribution with mean equal to four persons per day.(a) What is the probability that the number of people entering the intensive care unit on a particular day is two? (Round your answer to three decimal places.)[Input box for answer]What is the probability that the number of people entering the intensive care unit on a particular day is less than or equal to two? (Round your answer to three decimal places.)[Input box for answer](b) Is it likely that, on a given day, the number of people entering the intensive care unit, will exceed an \( x \)-value of 8? Calculate the mean and standard deviation of the probability distribution to explain your answer.- [ ] Yes, it is likely because \( x \) is less than 2 standard deviations above the mean.- [ ] No, it is not likely because \( x \) is more than 2 standard deviations above the mean.

Answered: The number x of people entering the…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

Need help on a and b

The number \( x \) of people entering the intensive care unit at a particular hospital on any one day has a Poisson probability distribution with mean equal to four persons per day.

(a) What is the probability that the number of people entering the intensive care unit on a particular day is two? (Round your answer to three decimal places.)

[Input box for answer]

What is the probability that the number of people entering the intensive care unit on a particular day is less than or equal to two? (Round your answer to three decimal places.)

[Input box for answer]

(b) Is it likely that, on a given day, the number of people entering the intensive care unit, will exceed an \( x \)-value of 8? Calculate the mean and standard deviation of the probability distribution to explain your answer.

- [ ] Yes, it is likely because \( x \) is less than 2 standard deviations above the mean.
- [ ] No, it is not likely because \( x \) is more than 2 standard deviations above the mean.

Expert Solution

Step 1: part a

(a) The number of people entering the intensive care unit on a particular day follows a Poisson distribution with a mean (λ) equal to four persons per day. To calculate the probability:

Probability that x = 2:
P(x = 2) = (e^(-λ) * λ^x) / x!
P(x = 2) = (e^(-4) * 4^2) / 2!
P(x = 2) ≈ (0.01832 * 16) / 2 ≈ 0.1832 (rounded to three decimal places).
Probability that x ≤ 2:

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

To calculate P(x = 0):

P(x = 0) = (e^(-λ) * λ^0) / 0!

P(x = 0) = (e^(-4) * 1) / 1 = 0.01832 (rounded to three decimal places).

To calculate P(x = 1):

P(x = 1) = (e^(-λ) * λ^1) / 1!

P(x = 1) = (e^(-4) * 4) / 1 = 0.1465 (rounded to three decimal places).

Now, add these probabilities:

P(x ≤ 2) = 0.01832 + 0.1465 + 0.1832 ≈ 0.348 (rounded to three decimal places)