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

Answer: (A)

Verified step by step guidance

1. Recall that a Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space, with the events happening independently and at a constant average rate \( \lambda \).
2. The probability mass function (PMF) for a Poisson random variable \( X \) with parameter \( \lambda \) is given by the formula for \( P(X = k) \), where \( k \) is a non-negative integer (\( k = 0, 1, 2, \ldots \)).
3. The PMF formula is derived from the limit of the binomial distribution and is expressed as: \[ P(X = k) = \frac{\lambda^{k} e^{-\lambda}}{k!} \]
4. Here, \( \lambda^{k} \) represents the rate parameter raised to the power of the number of events, \( e^{-\lambda} \) accounts for the probability of zero events occurring, and \( k! \) (k factorial) normalizes the count of events.
5. Compare the given options to this formula and identify the one that matches exactly the expression \( \frac{\lambda^{k} e^{-\lambda}}{k!} \) as the correct probability that \( X = k \) for a Poisson distribution.