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

Answer: (C)

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: \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \] where \( k \) is a non-negative integer (\( k = 0, 1, 2, \ldots \)), \( \lambda > 0 \), and \( e \) is the base of the natural logarithm.
3. Analyze each given expression by comparing it to the PMF formula. The correct expression must have: - \( \lambda^k \) in the numerator, - multiplied by \( e^{-\lambda} \), - and divided by \( k! \) in the denominator.
4. Check the first option: it has \( \frac{\lambda^k}{k} \), which is missing the factorial in the denominator and the exponential term, so it is incorrect.
5. Check the second and fourth options: they have incorrect powers of \( e \) and \( \lambda \) that do not match the PMF formula. The third option matches the PMF exactly, so it is the correct expression for \( P(X = k) \).