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

Answer: (D)

Verified step by step guidance

1. Recall that a Poisson distribution with parameter \( \lambda \) models the number of events occurring in a fixed interval, where 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. \[ P(X = k) = \frac{\lambda^{k}}{k!} e^{-\lambda} \]
4. Here, \( \lambda^{k} \) represents the parameter raised to the power of the number of events \( k \), \( k! \) is the factorial of \( k \), and \( e^{-\lambda} \) is the exponential term accounting for the probability of no events occurring beyond the observed count.
5. Compare each given expression to this formula to identify the correct one: the correct PMF must have \( \lambda^{k} \) in the numerator, \( k! \) in the denominator, and the exponential term must be \( e^{-\lambda} \).