A Poisson distribution with  λ =5.8 λ =5.8 and 4 < x < 74 < x < 7.

Use the probability distribution identified above to calculate the following:

**a.** The probability P(x) for the indicated values of x.
P(4)=P(4)= 

Round to 3 significant digits

**b.** The mean and standard deviation of the distribution.
 Mean (μ) = Mean (μ) = 
 SD (σ) = SD (σ) = 

Round to 3 significant digits

In this problem, we are dealing with a Poisson distribution with parameter \( \lambda = 5.8 \). Let's solve it step by step.

### a. Calculate \( P(4) \)

The probability mass function (PMF) for a Poisson distribution is given by the formula:

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

For \( x = 4 \), we will substitute \( \lambda = 5.8 \) and \( x = 4 \) into the formula:

\[
P(4) = \frac{5.8^4 \cdot e^{-5.8}}{4!}
\]

Let's calculate this.

### b. Mean and Standard Deviation

For a Poisson distribution:

- The **mean** \( \mu \) is equal to \( \lambda \).
\[
\mu = \lambda = 5.8
\]

- The **standard deviation** \( \sigma \) is the square root of \( \lambda \).
\[
\sigma = \sqrt{\lambda} = \sqrt{5.8}
\]

I'll calculate both the probability and standard deviation now.### a. The probability \( P(4) \) is approximately:

\[
P(4) \approx 0.143
\]

### b. The mean and standard deviation of the distribution are:

\[
\text{Mean} (\mu) = 5.8
\]
\[
\text{Standard Deviation} (\sigma) \approx 2.41
\]

Would you like further details or have any other questions?

Here are some related questions to consider:
1. What is \( P(5) \) for the same distribution?
2. How does the Poisson distribution change if \( \lambda \) is increased or decreased?
3. Can we calculate \( P(6) \) for this distribution?
4. How is the Poisson distribution used in real-life scenarios?
5. What is the cumulative probability \( P(x \leq 4) \)?

**Tip:** For Poisson distributions, the variance is equal to the mean, which is an important property.

A Poisson distribution with λ = 5.8 and 4 < x < 7. Use the probability distribution identified above to calculate the following: a. The probability P(x) for the indicated values of x. P(4) = ? (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 solve a Poisson distribution problem with λ = 5.8. Calculate the probability P(4), mean (μ), and standard deviation (σ) with detailed step-by-step solutions. Suitable for advanced high school and college-level students.

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