In Exercises 5–8, find the indicated probability using the Poisson distribution.


P(3) when μ = 6

Verified step by step guidance

1. Understand the Poisson distribution formula: P(x; μ) = (e^(-μ) * μ^x) / x!, where x is the number of occurrences, μ is the mean number of occurrences, and e is the base of the natural logarithm (approximately 2.718).
2. Identify the given values in the problem: x = 3 (the number of occurrences) and μ = 6 (the mean number of occurrences).
3. Substitute the given values into the formula: P(3; 6) = (e^(-6) * 6^3) / 3!.
4. Simplify the numerator: Calculate e^(-6), then multiply it by 6^3 (6 raised to the power of 3).
5. Simplify the denominator: Calculate 3! (3 factorial, which is 3 * 2 * 1), and then divide the numerator by this value to find the probability.

Key Concepts

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event. It is particularly useful for modeling rare events.

Parameter (μ)

In the context of the Poisson distribution, the parameter μ (mu) represents the average number of events in the specified interval. It is a crucial component for calculating probabilities, as it defines the shape and scale of the distribution. For example, if μ = 6, it indicates that, on average, 6 events are expected to occur.

Probability Mass Function (PMF)

The Probability Mass Function (PMF) of a discrete random variable gives the probability that the variable is equal to a specific value. For the Poisson distribution, the PMF is calculated using the formula P(X = k) = (e^(-μ) * μ^k) / k!, where k is the number of events, e is Euler's number, and k! is the factorial of k. This formula allows us to find the probability of observing exactly k events.