Given that $X$ has a Poisson distribution with parameter $\lambda$, which of the following is the correct expression for the probability that $X$ equals $k$?

• A. $\frac{{\lambda }^k}{e^k}$
• B. $\frac{{\lambda }^k{e}^{-\lambda }}{k!}$
• C. $\frac{{\lambda }^k}{k}$
• D. $\frac{e^{\lambda }}{k!}$

Answer: (B)

Verified step by step guidance

1. Recall that a Poisson distribution with parameter \( \lambda \) models the number of events occurring in a fixed interval of time or space, where these events happen independently and at a constant average rate \( \lambda \).
2. The probability mass function (PMF) for a Poisson random variable \( X \) giving the probability that \( X = k \) (where \( k \) is a non-negative integer) is given by the formula:
3. \[ P(X = k) = \frac{\lambda^{k} e^{-\lambda}}{k!} \]
4. This formula comes from the fact that the number of events follows a Poisson process, and the term \( \lambda^{k} \) represents the rate raised to the number of events, \( e^{-\lambda} \) accounts for the probability of no events occurring outside those counted, and \( k! \) normalizes for the number of ways to arrange these events.
5. Compare the given options to this formula to identify the correct expression for \( P(X = k) \). The correct choice must include all three components: \( \lambda^{k} \), \( e^{-\lambda} \), and division by \( k! \).