Question

Let x have a poisson distribution with a variance of 3. find p(x = 2).

Answer 1

The probability that a Poisson random variable X equals 2, given that its variance is 3, is approximately 0.2240.

Step-by-step explanation:

The student asks about finding the probability that a Poisson random variable X takes on a value exactly equal to 2, given that the variance of X is 3. We know for a Poisson distribution, the variance is equal to its mean, which is denoted by λ (lambda). Therefore, λ = 3.

Using the formula for the Poisson probability mass function (PMF), P(X = k) = (e^(-λ) * λ^k) / k!, we can calculate P(X = 2). Substituting λ = 3 and k = 2, we have:

P(X = 2) = (e^(-3) * 3^2) / 2! = (0.0497871 * 9) / 2 = 0.224041.

We can round this probability to four decimal places:

P(X = 2) ≈ 0.2240.

Answer 2

Since the variance of a Poisson distribution with rate parameter
`\lambda` is
`\lambda^2`, we know that
`\lambda=\sqrt3`. So,


`\mathbb P(X=2)=(e^(-\sqrt3)(\sqrt3)^2)/(2!)=\frac3{2e^(\sqrt3)}\approx0.2654`