If X is a Poisson random variable with a mean of 5, find:
4.1 P( X=3)
4.2 P( X≤3)
4.3 P( X>3)
Using the Poisson probability formula:
For a Poisson random variable X with mean λ:
1. P(X = k) = (e^(-λ) * λ^k) / k!
Given that mean (λ) = 5:
4.1 P(X = 3) = (e^(-5) * 5^3) / 3!
= (e^(-5) * 125) / 6
≈ 4.067%
To calculate the other probabilities, we will need to use the Poisson cumulative distribution function:
2. P(X ≤ k) = ∑[i=0 to k] (e^(-λ) * λ^i) / i!
3. P(X > k) = 1 - P(X ≤ k)
4.2 P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
= [(e^(-5) * 5^0) / 0!) + (e^(-5) * 5^1) / 1!) + (e^(-5) * 5^2) / 2!) + (e^(-5) * 5^3) / 3!]
= [e^(-5) + (e^(-5) * 5) + (e^(-5) * 25/2) + (e^(-5) * 125/6)]
≈ 26.512%
4.3 P(X > 3) = 1 - P(X ≤ 3)
= 1 - 0.26512
≈ 73.488%