Problem 25 - Basic Statistics for Business & Economics

Question

In a Poisson distribution, \(\mu=0.4\). a. What is the probability that \(x=0 ?\) b. What is the probability that \(x>0 ?\)

Step-by-Step Solution

Answer

a. 0.67032 b. 0.32968

Step 1: Understand the Poisson Distribution

The Poisson distribution is used for counting the number of events that happen in a fixed interval of time or space. The probability of observing exactly \( x \) events is given by the formula: \[ P(X = x) = \frac{{e^{-\mu} \mu^x}}{x!} \]where \( \mu \) is the average number of occurrences in the interval, \( x \) is the number of occurrences, and \( e \) is approximately 2.71828.

Step 2: Calculate Probability when x=0

To find \( P(X = 0) \) with \( \mu = 0.4 \):Use the formula: \[ P(X = 0) = \frac{{e^{-0.4} \times 0.4^0}}{0!} \]This simplifies to:\[ P(X = 0) = e^{-0.4} \times 1 \]Calculate \( e^{-0.4} \):\[ e^{-0.4} \approx 0.67032 \]Thus, \[ P(X = 0) \approx 0.67032 \].

Step 3: Calculate Probability when x > 0

The probability that \( X > 0 \) is complementary to \( X = 0 \). Thus, \[ P(X > 0) = 1 - P(X = 0) \]Substitute \( P(X = 0) \) from Step 2:\[ P(X > 0) = 1 - 0.67032 \]Calculate the result:\[ P(X > 0) \approx 0.32968 \].

Key Concepts

Probability Calculation

In statistics, understanding how to calculate probabilities is a crucial skill. For the Poisson distribution, which models the number of events in a fixed interval, you can calculate the probability of a specific event using the formula \[ P(X = x) = \frac{{e^{-\mu} \mu^x}}{x!} \]Here's a breakdown to ensure clarity:

• **\( e \)** is a constant approximately equal to 2.71828, representing the base of the natural logarithm.
• **\( \mu \)** is the mean number of occurrences in the time interval.
• **\( x \)** represents the actual number of events you want to find the probability for.
• **\( x! \)** is the factorial of \( x \), which is the product of all positive integers up to \( x \).

For example, if you want to calculate the probability of no events happening (\( x = 0 \)), substitute these values into the formula to get a numerical result. Probability calculations are essential for determining the likelihood of various outcomes, helping you to better understand random phenomena.

Random Events

Random events are occurrences that happen without a predictable pattern. In a Poisson distribution, these events could be customers arriving at a store or rainfall on specific days. A key property of random events in Poisson processes is that they are independent, meaning the occurrence of one event does not influence another.

• Randomness means each event's outcome is uncertain before it happens.
• In the context of Poisson distribution, the events occur within a fixed time frame or spatial area.
• Understanding randomness helps in calculating expected outcomes and can guide decision-making in uncertain environments.

A real-world example: consider a baker who receives calls for cake orders. These calls are random events that do not affect one another, making the Poisson distribution a suitable model to predict the number of calls on a given day. This concept of randomness is central to many statistical distributions, enabling the modeling and interpretation of data in an unpredictable world.

Statistical Distribution

A statistical distribution describes how probabilities are assigned to different possible outcomes. The Poisson distribution is a specific type of statistical distribution applied when you are counting occurrences in a fixed interval.

• It is defined by the average rate \( \mu \) of occurrences.
• Each event must be independent and occur with a constant probability over the chosen time span.

You can visualize a statistical distribution as a graph, where the x-axis represents possible outcomes and the y-axis their corresponding probabilities. In a Poisson distribution:

• The mean and variance are equal to \( \mu \).
• The distribution skews toward the right as \( \mu \) increases, indicating more occurrences are likely.

Statistical distributions like Poisson help in understanding the typical behavior of a random process, allowing you to predict possibilities and calibrate your expectations, which is crucial in data analysis and scientific research.