Problem 78 - Calculus for Biology and Medicine

Question

Suppose \(X\) is Poisson distributed with parameter \(\lambda=0.5\). Find \(P(X=k)\) for \(k=0,1,2\), and 3 .

Step-by-Step Solution

Answer

Calculate the probabilities using the formula: \( P(X=k) = \frac{e^{-0.5} \lambda^k}{k!} \) for \(k=0,1,2,3\).

Step 1: Understand the Poisson Probability Formula

The Poisson distribution is used to model the number of events that occur within a fixed interval of time or space. The probability of observing \(k\) events in an interval is given by the formula: \[ P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} \] where \(\lambda\) is the average rate of occurrence.

Step 2: Compute for k=0

Substitute \(k=0\) and \(\lambda=0.5\) into the formula: \[ P(X=0) = \frac{e^{-0.5} \cdot 0.5^0}{0!} = e^{-0.5} \cdot 1 = e^{-0.5} \] Calculate \( e^{-0.5} \) to get the probability.

Step 3: Compute for k=1

Substitute \(k=1\) and \(\lambda=0.5\) into the formula: \[ P(X=1) = \frac{e^{-0.5} \cdot 0.5^1}{1!} = e^{-0.5} \cdot 0.5 \] Calculate the value of \( e^{-0.5} \cdot 0.5 \) to find the probability.

Step 4: Compute for k=2

Substitute \(k=2\) and \(\lambda=0.5\) into the formula: \[ P(X=2) = \frac{e^{-0.5} \cdot 0.5^2}{2!} = \frac{e^{-0.5} \cdot 0.25}{2} \] Simplify the expression to find the probability.

Step 5: Compute for k=3

Substitute \(k=3\) and \(\lambda=0.5\) into the formula: \[ P(X=3) = \frac{e^{-0.5} \cdot 0.5^3}{3!} = \frac{e^{-0.5} \cdot 0.125}{6} \] Simplify the expression to determine the probability.

Step 6: Summarize Probabilities

The calculated probabilities are: \( P(X=0) = e^{-0.5} \), \( P(X=1) = e^{-0.5} \cdot 0.5 \), \( P(X=2) = \frac{e^{-0.5} \cdot 0.25}{2} \), and \( P(X=3) = \frac{e^{-0.5} \cdot 0.125}{6} \). These provide the respective probabilities of observing 0, 1, 2, and 3 events.

Key Concepts

Probability Formula

In understanding the Poisson Distribution, the probability formula is a cornerstone. This formula allows us to calculate the likelihood of a number of events occurring within a given period. It is given by

• The expression: \[ P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} \]
• Where \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
• \( \lambda \) is the parameter representing the average rate of occurrence.
• \( k \) stands for the number of events.
• \( k! \) (k-factorial) means we multiply all whole numbers from 1 to \( k \).

Use this formula to pinpoint how likely or unlikely certain occurrences are under the given conditions. For example, the formula is key when determining probabilities of events in a Poisson Distribution scenario.

Probability Calculation

To perform a probability calculation using the Poisson formula, follow a step-by-step method. This ensures clarity and accuracy in solving problems involving discrete random variables:

• Firstly, identify the average rate of occurrence \( \lambda \). For instance, \( \lambda=0.5 \).
• Next, determine the number of occurrences you wish to calculate for, denoted as \( k \). This can be 0, 1, 2, or any other non-negative integer.
• Substitute these values into the formula: - For \( k=0 \), \[ P(X=0) = \frac{e^{-0.5} \cdot 0.5^0}{0!} \] - Make similar substitutions for other values of \( k \).
• Simplify the expression to compute each probability.

This approach makes it easy to understand how each calculation is made stepwise, ensuring you grasp the underlying principles of probability calculation within the Poisson framework.

Discrete Distribution

Poisson Distribution is a prime example of a discrete distribution. It focuses on counting events, making it very different from continuous distributions which deal with measurements. Let’s break down why this concept is essential:

• Unlike continuous distributions where the values are uncountably infinite, discrete distributions work with countable values.
• For Poisson Distribution, discrete refers to counting distinct events like the number of emails received in an hour.
• Each probability, obtained using the Poisson probability formula, represents the likelihood of a specific number of these events.
• Ultimately, this distribution helps model phenomena where these discrete occurrences can vary widely between intervals.

Understanding this helps frame how probabilities are calculated within a perceived set of outcomes, thus emphasizing the count of occurrences.

Average Rate of Occurrence

The average rate of occurrence, denoted as \( \lambda \), is the expected number of times an event happens within a specific interval. In the context of the Poisson Distribution:

• The value \( \lambda \) forms the backbone of the distribution, greatly impacting the calculations.
• This parameter not only defines the frequency but also determines the distribution's shape.
• A smaller \( \lambda \) leads to fewer expected occurrences, whereas a larger \( \lambda \) suggests a higher frequency of events.
• For instance, with \( \lambda=0.5 \), we expect, on average, 0.5 occurrences within each interval.

Recognizing how \( \lambda \) functions can aid in visualizing and understanding how the randomness of events unfolds over time given various average rates.