Math Problem Statement

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

Solution

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

Part (a) - Probability for the indicated values of xx:

The probability mass function (PMF) of a Poisson distribution is given by:

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

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

P(3)=5.73e5.73!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 μ\mu is equal to λ\lambda, so: μ=λ=5.7\mu = \lambda = 5.7

  • The standard deviation σ\sigma is equal to λ\sqrt{\lambda}, so: σ=5.7\sigma = \sqrt{5.7}

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

Part (b) - Mean and Standard Deviation:

  • Mean μ=5.7\mu = 5.7
  • Standard Deviation σ=2.387\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)P(x) for values other than x=3x = 3?
  2. What is the cumulative probability P(3x6)P(3 \leq x \leq 6)?
  3. How does the Poisson distribution differ from the binomial distribution?
  4. How would the probabilities change if λ\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., Var(X)=λ\text{Var}(X) = \lambda.

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Math Problem Analysis

Mathematical Concepts

Probability Theory
Poisson Distribution

Formulas

P(x) = (λ^x * e^(-λ)) / x!
Mean (μ) = λ
Standard Deviation (σ) = √λ

Theorems

Poisson Theorem

Suitable Grade Level

Undergraduate Level (Introductory Statistics)