Probability Distribution of a Poisson Random Variable

The Poisson distribution is a discrete probability distribution that is often used to model the number of events that occur in a fixed interval of time or space. It is defined by a single parameter, λ (lambda), which represents the average rate of occurrence of the events.

The probability mass function (PMF) of a Poisson random variable X is given by:

P(X = k) = (e^(-λ) * λ^k) / k!

Where e is the base of the natural logarithm and k! is the factorial of k.

Given Information

In this question, we are given that the probability of X being equal to 1 (P(X = 1)) is equal to the probability of X being equal to 2 (P(X = 2)).

Using the Poisson PMF

To find the value of P(X = 3), we can use the Poisson PMF. Let's denote P(X = 1) as p.

Given that P(X = 1) = P(X = 2), we can write:

p = (e^(-λ) * λ^1) / 1!
p = (e^(-λ) * λ) / 1

We can rearrange this equation to solve for λ:

λ = p * e^(λ)

Substituting this value of λ back into the PMF, we can find P(X = 3):

P(X = 3) = (e^(-λ) * λ^3) / 3!

Substituting the value of λ from the previous equation, we get:

P(X = 3) = (e^(-p * e^(λ)) * (p * e^(λ))^3) / 3!

Simplifying this expression, we obtain:

P(X = 3) = (e^(-p * e^(p * e^(λ))) * (p * e^(p * e^(λ)))^3) / 3!

Numerical Calculation

To find the numerical value of P(X = 3), we need to know the value of p. Unfortunately, the value of p is not provided in the question. Hence, without the value of p, we cannot determine the exact value of P(X = 3).

However, the correct answer is given as 0.18, which suggests that a specific value of p has been assumed. Without additional information, we cannot determine the exact value of p that corresponds to P(X = 3) = 0.18.

Hence, we cannot explain the answer of 0.18 without further information or assumptions.