Final answer:
The probability that a random variable X, which has a Poisson distribution with a mean of 4, takes a value of 5, is approximately 0.156. This is computed using the Poisson probability mass function with the given mean and desired value.
Step-by-step explanation:
The question deals with the Poisson distribution, which is a probability distribution that describes the likelihood of a given number of events occurring in a fixed interval of time or space, assuming that these events occur with a known constant mean rate and independently of the time since the last event. The mean (μ or λ) of the distribution is given as 4. Therefore, if X follows a Poisson distribution with a mean of 4, the probability of observing exactly 5 events (P(X = 5)) can be calculated using the Poisson probability mass function (pmf):
P(X = x) = (μ^x e^-μ) / x!
Substituting the given values:
P(X = 5) = (4^5 * e^-4) / 5! = 1024 * e^-4 / 120 = 0.156/15
This computation gives the probability as approximately 0.156, which corresponds to answer choice b.