Problem 3 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Let \(N\) have a Pois (4) distribution. What is \(\mathrm{P}(N=4) ?\)

Step-by-Step Solution

Answer

\( P(N = 4) \approx 0.1954 \)

Step 1: Understanding the Poisson distribution

A Poisson distribution with parameter \(\lambda\) is used to model the number of events occurring within a given time period. Here, \(\lambda = 4\), which means the average number of occurrences in the period is 4.

Step 2: Apply the Poisson probability formula

The probability of observing \(k\) events in a Poisson distribution is given by the formula \[ P(N = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]For this problem, \(k = 4\) and \(\lambda = 4\). Substitute these values into the formula to get \[ P(N = 4) = \frac{e^{-4} \cdot 4^4}{4!} \].

Step 3: Calculate exponent and factorial

First, calculate \(e^{-4}\), which is approximately \0.018316\. Then, calculate \(4^4 = 256\) and \(4! = 24\).

Step 4: Compute the probability

Substitute the calculated values back into the formula: ~\[ P(N = 4) = \frac{0.018316 \cdot 256}{24} \].

Step 5: Finalize the calculation

Perform the multiplication: \(0.018316 \times 256 = 4.690496\), then divide by 24 to find \[ P(N = 4) = \frac{4.690496}{24} \approx 0.1954 \].

Key Concepts

Probability Calculation

When we talk about probability calculation in relation to the Poisson distribution, we aim to determine the likelihood of a specific number of events occurring within a fixed interval. In this context, probability indicates how often you should expect something to happen.

To calculate probability using the Poisson distribution:

• Identify the average rate of occurrence (\( \lambda \) in this case).
• Determine the specific number of events of interest (\( k \)).
• Insert these values into the Poisson probability formula.

These steps allow you to calculate the chance of exactly \( k \) events occurring when events happen independently and at a constant average rate.

For our example, we calculated the probability of exactly 4 events happening when the average rate of occurrence is 4. This probability provides significant insights, especially in fields like telecommunications, biology, and retail, where events are random yet follow an observable average pattern.

Poisson Probability Formula

The Poisson probability formula is a fundamental tool in probability theory which helps model the number of events occurring within a fixed interval. The formula is given by:\[ P(N = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]Here,

• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
• \( \lambda \) represents the average event rate.
• \( k \) is the number of events we are interested in.
• \(! \) denotes factorial, an operation we'll delve into next.

The magic of the formula lies in balancing the exponential function and the power of the average, divided by a factorial. This sets up the shape of the probability distribution, perfectly capturing the randomness with predictable averages.

In the exercise, inserting \( \lambda = 4 \) and \( k = 4 \) allows us to compute this probability accurately.

Factorial Calculation

Understanding factorial calculation is crucial when working with the Poisson probability formula. A factorial, denoted by \( n! \), is the product of all positive integers up to a number \( n \). It's particularly useful in permutations and combinations, showing up in various statistical formulas.

For example, the factorial of 4, written as \( 4! \), is:\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]Factorials grow very quickly as \( n \) increases, which impacts how probabilities scale in the Poisson formula. This scaling makes it possible to precisely model rare events where only small numbers of events occur.

In our problem, calculating \( 4! \) was essential to finding \( P(N=4) \). It allowed us to normalize the probability by accounting for the number of possible arrangements of \( k \) events.