Question

Consider a Poisson probability distribution with = 1. Determine the probability of P(x 4). Round your final answer to four decimal places, and include the zero before the decimal place. E.g., 0.1234

Answer 1

The probability of P(x ≤ 4) in a Poisson distribution with a mean of 1 is approximately 0.6288 when rounded to four decimal places.

Step-by-step explanation:

To calculate the probability that \(x\) is less than or equal to 4 in a Poisson distribution with a mean (\(μ\)) of 1, we employ the Poisson probability function. The formula for the Poisson probability function is given by \(P(x; μ) = \frac{e^{-μ} * μ^x}{x!}\). Here, \(e\) is Euler's number, approximately equal to 2.71828.

For this specific scenario, we want to find \(P(X \leq 4)\), which involves summing the probabilities for \(x = 0, 1, 2, 3,\) and \(4\). Using technology or tables that compute Poisson probabilities, we find:

\[P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) \approx 0.6288.\]

Therefore, the probability that \(x\) is less than or equal to 4 in this Poisson distribution is approximately 0.6288 when rounded to four decimal places.

Understanding and utilizing probability distributions, such as the Poisson distribution, is essential in various fields, including statistics, finance, and science. These distributions help quantify the likelihood of different outcomes, enabling informed decision-making and analysis in a wide range of applications.