Problem 26 - Basic Statistics for Business & Economics

Question

In a Poisson distribution, \(\mu=4\) a. What is the probability that \(x=2 ?\) b. What is the probability that \(x \leq 2 ?\) c. What is the probability that \(x>2 ?\)

Step-by-Step Solution

Answer

a: 0.1465, b: 0.2381, c: 0.7619

Step 1: Understanding Poisson Distribution

A Poisson distribution is used to model the number of events occurring within a fixed interval of time or space. It is defined by the mean rate \( \mu \), which in this case is given as 4. The probability mass function (PMF) of the Poisson distribution for a value \( x \) is given by: \[ P(X = x) = \frac{e^{-\mu} \mu^x}{x!} \] where \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Step 2: Calculating Probability for x = 2

Using the PMF of the Poisson distribution, calculate the probability that \( x = 2 \): \[ P(X = 2) = \frac{e^{-4} \cdot 4^2}{2!} \] First, compute \( e^{-4} \) and \( 4^2 = 16 \). Then compute \( 2! = 2 \). Ultimately, \[ P(X = 2) = \frac{0.0183156 \cdot 16}{2} = \frac{0.2930496}{2} = 0.146525 \] Hence, the probability \( P(X = 2) \) is approximately 0.1465.

Step 3: Calculating Probability for x ≤ 2

To find \( P(X \leq 2) \), sum up the probabilities of \( X = 0, 1, \) and \( 2 \). \[ P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) \] Calculate each: - \( P(X = 0) = \frac{e^{-4} \cdot 4^0}{0!} = 0.0183156 \) - \( P(X = 1) = \frac{e^{-4} \cdot 4^1}{1!} = 0.0732624 \) - Use \( P(X = 2) \) from Step 2: 0.146525 Sum them to get: \[ P(X \leq 2) = 0.0183156 + 0.0732624 + 0.146525 = 0.238103 \] Thus, \( P(X \leq 2) \) is approximately 0.2381.

Step 4: Calculating Probability for x > 2

The probability \( P(X > 2) \) is the complement of \( P(X \leq 2) \), so: \[ P(X > 2) = 1 - P(X \leq 2) \] From Step 3, \( P(X \leq 2) \) is 0.238103, thus: \[ P(X > 2) = 1 - 0.238103 = 0.761897 \] Hence, the probability \( P(X > 2) \) is approximately 0.7619.

Key Concepts

Probability Mass Function

The Probability Mass Function (PMF) is a fundamental concept in probability theory and statistics, especially when dealing with discrete random variables like in a Poisson distribution. For a Poisson distribution, the PMF describes the probability of observing a certain number of events, \( x \), in a fixed interval. This is expressed mathematically by: \[ P(X = x) = \frac{e^{-\mu} \mu^x}{x!} \] where \( e \) is the Euler's number (approximately 2.71828), \( \mu \) is the average number of events, and \( x! \) denotes the factorial of \( x \).In our context, with \( \mu = 4 \), if we're calculating for \( x = 2 \), we plug into the formula to find the exact probability of exactly 2 events occurring. The PMF helps to solve such problems by providing a systematic way to calculate these probabilities plainly and accurately.

Poisson Probability Calculation

Calculating probabilities using the Poisson distribution involves using the PMF and involves a few straightforward steps. For example, to find \( P(X = 2) \) where \( \mu = 4 \), you substitute into the PMF as follows:

• Compute \( e^{-4} \), which is roughly 0.0183156.
• Calculate \( 4^2 \), which results in 16.
• Factorial of 2 (2!), which equates to 2.

Then, substitute these values into the PMF:\[ P(X = 2) = \frac{0.0183156 \cdot 16}{2} = 0.146525 \]This result shows that there is approximately a 14.65% chance of precisely 2 events occurring in this Poisson framework. This method is repeated similarly for other values of \( x \), making Poisson calculations methodical and efficient.

Complement Rule

The complement rule is a helpful tool in probability, particularly when calculating the probability of a statement that can easily be derived from the complement of a simpler statement. In the Poisson distribution problem, the complement rule helps in finding \( P(X > 2) \) by using \( P(X \leq 2) \). Given that the total probability must sum to 1, the probability of \( X > 2 \) is calculated as:\[ P(X > 2) = 1 - P(X \leq 2) \]Using the data, if \( P(X \leq 2) = 0.238103 \), then:\[ P(X > 2) = 1 - 0.238103 = 0.761897 \]This implies a 76.19% chance of more than 2 events occurring. The complement rule simplifies computations by requiring you to calculate only the simpler probability and subtracting it from one to find its complement, the desired probability.