Problem 26 - Basic Statistics for Business and Economics with Student

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. \(P(X=2) \approx 0.1465\); b. \(P(X \leq 2) \approx 0.2381\); c. \(P(X>2) \approx 0.7619\).

Step 1: Understanding Poisson Distribution

The Poisson distribution models the probability of a given number of events happening in a fixed interval of time or space. The parameter \(\mu\) represents the mean number of events in this interval. It is defined by the probability mass function: \(P(X=k) = \frac{e^{-\mu} \mu^k}{k!}\).

Step 2: Calculate Probability for x=2

Use the probability mass function to find \(P(X=2)\). Substitute \(\mu = 4\) and \(k = 2\) into the formula: \(P(X=2) = \frac{e^{-4} \cdot 4^2}{2!} = \frac{e^{-4} \cdot 16}{2}\). Calculate this value using a calculator.

Step 3: Calculate Cumulative Probability for x <= 2

Find the sum of probabilities for \(x=0, 1, \text{and} \ 2\). Use the formula for each and sum them: \(P(X \leq 2) = P(X=0) + P(X=1) + P(X=2)\). The values are calculated as follows: \(P(X=0) = \frac{e^{-4} \cdot 4^0}{0!}, \ P(X=1) = \frac{e^{-4} \cdot 4^1}{1!}, \ P(X=2) = \frac{e^{-4} \cdot 16}{2}\). Add these probabilities together.

Step 4: Calculate Probability for x>2 Using Complement Rule

Use the complement rule to find \(P(X>2) = 1 - P(X \leq 2)\). Since you already calculated \(P(X \leq 2)\) in Step 3, subtract this value from 1 to find \(P(X>2)\).

Key Concepts

Probability Mass Function

The Poisson distribution is a probability distribution used to model the number of events occurring in a fixed interval of time or space. The key parameter here is \(\mu\), which represents the mean number of events expected in that interval.
The probability mass function (PMF) is essential in understanding how probabilities are assigned to each possible outcome. The Poisson PMF is defined as:
\[ P(X=k) = \frac{e^{-\mu} \mu^k}{k!} \]
Here, \(e\) is the base of the natural logarithm, approximately equal to 2.71828. \(k\) is the number of events occurring within the fixed interval, and \(k!\) (k factorial) counts the number of ways to arrange \(k\) events. This function allows you to calculate the precise probability that exactly \(k\) events occur.

• The PMF is specific to discrete random variables like \(X\) in a Poisson distribution.
• It helps directly compute the probability of specific outcomes, such as finding \(P(X=2)\) using \(\mu\) and \(k\).

By substituting \(\mu=4\) and \(k=2\) into the PMF formula, we see how it calculates \(P(X=2)\) and derives the outcome probability.

Complement Rule

In probability theory, the complement rule is a fundamental principle used to calculate the probability of the complement of an event. It states that for any event \(A\), the probability of \(A\) not occurring (the complement of \(A\)) is equal to 1 minus the probability of \(A\) occurring.
This is written as:
\[ P(A^c) = 1 - P(A) \]
Where \(A^c\) represents the complement of \(A\). When solving for \(P(X>2)\) in a Poisson distribution, you can use the complement rule by thinking of the event "\(X > 2\)" as the complement of "\(X \leq 2\)". Therefore, the probability that \(X > 2\) is calculated by subtracting \(P(X \leq 2)\) from 1. This method is useful because:

• It simplifies the calculation when the complement of an event is easier to find than the original event itself.
• It is often used in Poisson distribution problems to find probabilities like "more than" or "greater than" events, as it converts a more laborious cumulative calculation into a single step.

Thus, using the complement rule allows us to efficiently ascertain the likelihood of a complementary event occurring.

Cumulative Probability

Cumulative probability refers to the probability that a random variable is less than or equal to a certain value. In the context of the Poisson distribution, cumulative probability is used to calculate the likelihood of observing \(x\) or fewer events.
When calculating \(P(X \leq 2)\) for a Poisson distribution with \(\mu=4\), you need to sum the probabilities of all events from \(x=0\) to \(x=2\), using the PMF for each value:

• \(P(X=0)\)
• \(P(X=1)\)
• \(P(X=2)\)

After determining each of these individual probabilities, the cumulative probability is the sum \(P(X \leq 2) = P(X=0) + P(X=1) + P(X=2)\).
This cumulative approach is crucial because:

• It enables the determination of the probability of a random variable being less than or equal to a specified point, offering insights into the range of outcomes less than \(k\).
• It's particularly helpful in real-world applications where you need to consider all possible outcomes up to a certain event number.

Understanding cumulative probability is important for interpreting the likelihood of aggregated or sequentially accumulated outcomes in statistical scenarios.