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Chapter 4: Problem 6

Find the indicated probability using the Poisson distribution. $$ \text { Find } P(3) \text { when } \mu=6 $$

Short Answer

Expert verified
P(3; 6) 鈮� 0.08923

Step by step solution

01

Understand the Poisson Formula

The Poisson distribution is used to model the number of events occurring within a given time period. The probability of observing exactly \( k \) events is given by the formula: \[P(k; \mu) = \frac{e^{-\mu} \mu^k}{k!}\]where \( \mu \) is the average number of events, \( k \) is the number of occurrences, and \( e \) is the base of the natural logarithm (approximately 2.71828).
02

Identify Parameters

In this problem, we want to find \( P(3) \) with \( \mu = 6 \). This means we are looking for the probability of exactly 3 events occurring when the average rate \( \mu \) is 6.
03

Substitute the Values in the Formula

We need to substitute \( k = 3 \) and \( \mu = 6 \) into the Poisson formula:\[P(3; 6) = \frac{e^{-6} \, 6^3}{3!}\]
04

Calculate the Exponential and Power Components

Compute the components separately: \( e^{-6} \approx 0.002478752 \) and \( 6^3 = 216 \). Substitute these values into the formula: \[P(3; 6) = \frac{0.002478752 \times 216}{3!}\]
05

Calculate the Factorial

Now calculate the factorial: \( 3! = 3 \times 2 \times 1 = 6 \). Substitute this into the formula:\[P(3; 6) = \frac{0.002478752 \times 216}{6}\]
06

Compute the Probability

Finish the calculation by multiplying and dividing the components:\[P(3; 6) = \frac{0.5354}{6} \approx 0.08923\]
07

Interpret the Result

The probability of observing exactly 3 events when the average number of occurrences is 6 is approximately 0.08923.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Calculation
Probability calculation is an essential idea in understanding how likely an event is to occur. In the context of a Poisson distribution, probability calculation allows us to determine the likelihood of observing a specific number of events occurring within a predetermined time frame. For example, when asked to find the probability of exactly 3 events occurring given that the average number of occurrences, \( \mu \), is 6, the Poisson probability formula is employed.
  • The formula for Poisson probability is: \[P(k; \mu) = \frac{e^{-\mu} \mu^k}{k!},\] where:
  • \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
  • \( k \) denotes the number of events occurring.
  • \( \mu \) is the average number of events in the interval.
By substituting the values of \( \mu = 6 \) and \( k = 3 \) into this formula, and performing the calculations as shown in the solution, we gain insight into the probability of the event in question.
Exponential Function
The exponential function is crucial in the calculation and understanding of Poisson probabilities. Within the context of Poisson distribution, the term \( e^{-\mu} \) arises frequently.
  • Here, \( e \) is a constant, approximately 2.71828, and represents the base of natural logarithms.
  • When \( \mu = 6 \), the expression becomes \( e^{-6} \), which is the exponential decay corresponding to the average number of events.
The exponential function illustrates how the probability decreases as the number of desired events, \( k \), deviates from the average \( \mu \). In practice, calculating \( e^{-6} \) yields approximately 0.002478752. It forms the backbone of the probability calculation by adjusting the probability relative to the average rate of occurrence.
Factorial
Factorials are mathematical expressions multiplying a series of descending natural numbers. It is denoted by the symbol \(!\). In the Poisson probability formula, the factorial appears in the denominator.
  • The factorial of \( k \) is \( k! \), which is defined as \( k \times (k-1) \times (k-2) \times \cdots \times 1 \).
  • For instance, \( 3! = 3 \times 2 \times 1 = 6 \).
When incorporated into the Poisson formula \[P(k; \mu) = \frac{e^{-\mu} \mu^k}{k!},\] the factorial function helps to determine the weight of the probability for the occurrence of \( k \) events. Understanding how factorials operate ensures accurate computation of probabilities using the Poisson model.
Statistics
Statistics is the branch of mathematics that deals with collecting, analyzing, and interpreting data. The Poisson distribution is a statistical measure used to model events occurring at a given rate, where each event is independent of the others.
  • Poisson distribution helps in predicting probabilities of various numbers of events in fixed intervals of time or space.
  • This distribution is particularly useful when the data is concerning rare events or occurrences with a low probability.
By understanding statistical concepts such as the Poisson distribution, students can apply these methods to real-world problems to deduce probabilities and make data-driven predictions. This forms the fundamental backbone behind using advanced statistical models to inform decisions, enhance strategies, and make sense of random events.

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Most popular questions from this chapter

Defective Parts A spare parts seller finds that 3 in every 100 parts sold is defective. Find the probability that (a) the first defective part is the eighth part sold, (b) the first defective part is the first, second or third part sold, (c) none of the first 8 parts sold are defective.

Cheating Sixty-eight percent of undergraduate students admit to cheating on tests or in written work. You randomly select six undergraduate students. Find the probability that the number of undergraduate students who admit to cheating on tests or in written work is (a) exactly four, (b) more than two, and (c) at most five. (Source: The Atlantic)

In your own words, describe the difference between the value of \(x\) in a binomial distribution and in the Poisson distribution.

Find the indicated probability using the geometric distribution. $$ \text { Find } P(3) \text { when } p=0.65 $$

Find the indicated probability using the geometric distribution. $$ \text { Find } P(8) \text { when } p=0.28 \text {. } $$
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