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 , we can solve the questions as follows:
Part (a) - Probability for the indicated values of :
The probability mass function (PMF) of a Poisson distribution is given by:
Here, and for , we can plug into the formula:
Let me calculate this.
Part (b) - Mean and Standard Deviation:
For a Poisson distribution:
-
The mean is equal to , so:
-
The standard deviation is equal to , so:
I'll compute these values as well.### Part (a) - Probability :
Part (b) - Mean and Standard Deviation:
- Mean
- Standard Deviation
Let me know if you have any questions or need more details!
Here are 5 related questions to expand your understanding:
- How do we calculate the probability for values other than ?
- What is the cumulative probability ?
- How does the Poisson distribution differ from the binomial distribution?
- How would the probabilities change if were larger or smaller?
- What is the variance of the Poisson distribution?
Tip: In a Poisson distribution, the variance is always equal to the mean, i.e., .
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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)
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