What is the value of Prob(X = 3) for a Poisson distribution where Prob(X = 1) = Prob(X = 2)? (GATE ME 2022 Q47)

Question

Let a random variable X follow Poisson distribution such that Prob(X = 1) = Prob(X = 2). The value of Prob(X = 3) is __________ (round off to 2 decimal places).

Accepted Answer

To find Prob(X = 3) for a Poisson distribution, we first need to determine the parameter λ (lambda).
The probability mass function (PMF) for a Poisson distribution is: P(X=k) = (e^(-λ) * λ^k) / k!

Given: P(X = 1) = P(X = 2)
(e^(-λ) * λ^1) / 1! = (e^(-λ) * λ^2) / 2!
λ = λ^2 / 2
2λ = λ^2
λ^2 - 2λ = 0
λ(λ - 2) = 0
Since λ > 0 for a Poisson distribution, we take λ = 2.

Now, we need to find Prob(X = 3) with λ = 2:
P(X = 3) = (e^(-2) * 2^3) / 3!
P(X = 3) = (e^(-2) * 8) / (3 * 2 * 1)
P(X = 3) = (e^(-2) * 8) / 6
P(X = 3) = (e^(-2) * 4) / 3

Using e ≈ 2.71828, e^(-2) ≈ 0.135335
P(X = 3) ≈ (0.135335 * 4) / 3
P(X = 3) ≈ 0.54134 / 3
P(X = 3) ≈ 0.180446

Rounding off to 2 decimal places, P(X = 3) ≈ 0.18

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