Probability for Poisson Distributions

Question

Directions: Find each specified probability. Assume that the random variable X has a Poisson distribution with the given value of λ.

P (X < 5) , λ = 7.10

Answer 1

The Probability Mass Function for a Poisson random variable is

P(X = x) = (e^-λ)•(λ^x)/(x!), where λ is the expected value E(X).

This can be done by "brute force" by plugging in the appropriate values for each of the numbers from 0 through 5, inclusive. That is, we can add the following:

P(X = 0) + P(X = 1) + P(X = 3) + P(X = 4) + P(X = 5)

(e^-7.1)•(7.1⁰)/(0!) + (e^-7.1)•(7.1¹)/(1!) + (e^-7.1)•(7.1²)/(2!) + (e^-7.1)•(7.1³)/(3!) + (e^-7.1)•(7.1⁴)/(4!) + (e^-7.1)•(7.1⁵)/(5!) ≈ 0.28811944.

If you want to make the calculations a little easier, you can factor out e^-7.1, and the problem looks like this: e^-7.1[(7.1⁰)/(0!) + (7.1¹)/(1!) + (7.1²)/(2!) + (7.1³)/(3!) + (7.1⁴)/(4!) + (7.1⁵)/(5!)].

Even simpler is e^-7.1(1 + 7.1 + 7.1²/2 + 7.1³/3! + 7.1⁴/4! + 7.1⁴/4!)

If you have the technology available, you can use poissoncdf in the DISTR (2ND VARS) menu of the TI calculators. This will give you P(X < 5) ≈ 0.28811944.