Assume that *X* is a Poisson random variable with *μ* = 20. Calculate the following probabilities.

**Note: Do not round intermediate calculations. Round your final answers to 4 decimal places.                                          a**	﻿﻿﻿**P(X ≤ 11)**

**b**	﻿﻿﻿**P(X = 14)**

**c**	﻿﻿﻿**P(X > 18)**

**d**	﻿﻿﻿**P(19 ≤ X ≤ 27)**

In order to solve this, we'll use the properties of the Poisson distribution. For a Poisson random variable \( X \) with mean \( \mu \), the probability mass function (PMF) is given by:

\[
P(X = k) = \frac{\mu^k e^{-\mu}}{k!}
\]
where \( \mu = 20 \) and \( k \) is the desired value of \( X \).

We can also calculate cumulative probabilities using the cumulative distribution function (CDF) for the Poisson distribution.

### a) \( P(X \leq 11) \)
This is the cumulative probability that \( X \) takes on values from 0 to 11. We need to sum up the individual probabilities:

\[
P(X \leq 11) = \sum_{k=0}^{11} \frac{20^k e^{-20}}{k!}
\]

### b) \( P(X = 14) \)
This is the probability of \( X \) taking on the value 14:

\[
P(X = 14) = \frac{20^{14} e^{-20}}{14!}
\]

### c) \( P(X > 18) \)
This is the complement of the cumulative probability up to 18:

\[
P(X > 18) = 1 - P(X \leq 18)
\]
We need to sum the probabilities for \( X \leq 18 \) and subtract from 1.

### d) \( P(19 \leq X \leq 27) \)
This is the sum of the probabilities between 19 and 27:

\[
P(19 \leq X \leq 27) = \sum_{k=19}^{27} \frac{20^k e^{-20}}{k!}
\]

I will now calculate each of these probabilities using Python.Here are the calculated probabilities:

- **a) \( P(X \leq 11) = 0.0214 \)**
- **b) \( P(X = 14) = 0.0387 \)**
- **c) \( P(X > 18) = 0.6186 \)**
- **d) \( P(19 \leq X \leq 27) = 0.5661 \)**

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How would the probabilities change if \( \mu \) were 30 instead of 20?
2. What is the expected value and variance of \( X \) in the Poisson distribution?
3. How can the Poisson distribution be approximated using a normal distribution?
4. What is \( P(X = 0) \) for this Poisson distribution with \( \mu = 20 \)?
5. How would you calculate the cumulative probability for \( P(X \leq 15) \)?

**Tip:** When \( \mu \) is large, the Poisson distribution can be approximated using a normal distribution with mean \( \mu \) and variance \( \mu \).

Assume that X is a Poisson random variable with μ = 20. Calculate the following probabilities: a) P(X ≤ 11), b) P(X = 14), c) P(X > 18), d) P(19 ≤ X ≤ 27).

Learn how to calculate probabilities for a Poisson random variable X with mean μ = 20. This problem includes detailed calculations for P(X ≤ 11), P(X = 14), P(X > 18), and P(19 ≤ X ≤ 27) using the Poisson PMF and CDF. Suitable for advanced high school or undergraduate students.

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