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.