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

Answer: (A)

Verified step by step guidance

1. Recall that a Poisson distribution with parameter \( \lambda \) models the probability of a given number of events \( k \) occurring in a fixed interval of time or space, where these events happen with a known constant mean rate \( \lambda \) and independently of the time since the last event.
2. The probability mass function (PMF) for a Poisson random variable \( X \) is given by the formula for \( P(X = k) \), which expresses the probability that the random variable \( X \) takes the value \( k \).
3. The general form of the PMF for a Poisson distribution is: \[ P(X = k) = \frac{\lambda^{k} e^{-\lambda}}{k!} \] where: - \( \lambda^{k} \) represents the rate parameter raised to the power of \( k \), - \( e^{-\lambda} \) is the exponential decay factor, - \( k! \) is the factorial of \( k \), accounting for the number of ways to arrange \( k \) events.
4. Compare each given expression to this formula. The correct expression must have \( \lambda^{k} \) in the numerator, multiplied by \( e^{-\lambda} \), and divided by \( k! \).
5. Identify the expression that matches this structure exactly, confirming it as the correct PMF for the Poisson distribution.