A Poisson distribution with  λ =5.7
and 3 < x < 6
.

Use the probability distribution identified above to calculate the following:
a. The probability P(x) for the indicated values of x.
P(3)=

Round to 3 significant digits
b. The mean and standard deviation of the distribution.
Mean (μ) =

SD (σ) =

Round to 3 significant digits

Given the Poisson distribution with \( \lambda = 5.7 \), we can solve the questions as follows:

### Part (a) - Probability for the indicated values of \( x \):
The probability mass function (PMF) of a Poisson distribution is given by:

\[
P(x) = \frac{{\lambda^x e^{-\lambda}}}{{x!}}
\]

Here, \( \lambda = 5.7 \) and for \( P(3) \), we can plug \( x = 3 \) into the formula:

\[
P(3) = \frac{{5.7^3 e^{-5.7}}}{{3!}}
\]

Let me calculate this.

### Part (b) - Mean and Standard Deviation:
For a Poisson distribution:
- The **mean** \( \mu \) is equal to \( \lambda \), so:
  \[
  \mu = \lambda = 5.7
  \]

- The **standard deviation** \( \sigma \) is equal to \( \sqrt{\lambda} \), so:
  \[
  \sigma = \sqrt{5.7}
  \]

I'll compute these values as well.### Part (a) - Probability \( P(3) \):
\[
P(3) = 0.103
\]

### Part (b) - Mean and Standard Deviation:
- Mean \( \mu = 5.7 \)
- Standard Deviation \( \sigma = 2.387 \)

Let me know if you have any questions or need more details!

Here are 5 related questions to expand your understanding:
1. How do we calculate the probability \( P(x) \) for values other than \( x = 3 \)?
2. What is the cumulative probability \( P(3 \leq x \leq 6) \)?
3. How does the Poisson distribution differ from the binomial distribution?
4. How would the probabilities change if \( \lambda \) were larger or smaller?
5. What is the variance of the Poisson distribution?

**Tip:** In a Poisson distribution, the variance is always equal to the mean, i.e., \( \text{Var}(X) = \lambda \).

A Poisson distribution with λ = 5.7 and 3 < x < 6. Use the probability distribution identified above to calculate the following: a. The probability P(x) for the indicated values of x. P(3)= Round to 3 significant digits. b. The mean and standard deviation of the distribution. Mean (μ) = SD (σ) = Round to 3 significant digits.

Learn how to calculate P(3) using a Poisson distribution with λ = 5.7, and find the mean and standard deviation. The problem involves key probability concepts like the Poisson formula, mean, and standard deviation calculations. Suitable for students in introductory statistics courses.

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