If a random variable x follows a Poisson distribution such that Prob (x = 1) = Prob (x = 2), find the mean and variance. Also find Prob (x = 0).

Understanding the Poisson Distribution

The probability mass function (PMF) of a Poisson distribution is given by:

P(x = k) = (e^-λ * λ^k) / k!

where:

• x is the random variable representing the number of events
• k is a non-negative integer (0, 1, 2, ...)
• λ (lambda) is the average rate of events
• e is Euler's number (approximately 2.71828)
• k! is the factorial of k

Finding the Mean (λ) and Variance

We are given that Prob(x = 1) = Prob(x = 2). Using the PMF, we can write:

(e^-λ * λ¹) / 1! = (e^-λ * λ²) / 2!

Simplifying the equation:

λ = λ² / 2

Since λ cannot be zero (otherwise it wouldn't be a Poisson distribution), we can divide both sides by λ:

1 = λ / 2

Therefore, λ = 2

For a Poisson distribution, the mean (μ) is equal to the variance (σ²), and both are equal to λ.

Therefore, Mean (μ) = λ = 2

Variance (σ²) = λ = 2

Calculating Prob(x = 0)

Now, we need to find Prob(x = 0) using the PMF and the value of λ we found:

Prob(x = 0) = (e^-2 * 2⁰) / 0!

Since 2⁰ = 1 and 0! = 1:

Prob(x = 0) = e^-2

Prob(x = 0) ≈ 0.1353

Summary of Results

Parameter	Value
Mean (λ)	2
Variance (λ)	2
Prob(x = 0)	e-2 ≈ 0.1353