Derivation of Poisson Distribution as a Limit of Binomial Distribution - 19.X.3 | 19. Poisson Distribution | Mathematics - iii (Differential Calculus) - Vol 3
K12 Students

Academics

AI-Powered learning for Grades 8–12, aligned with major Indian and international curricula.

Academics

Grades

Grade 8 Grade 9 Grade 10 Grade 11 Grade 12

Curriculum

CBSE ICSE IB
Professionals

Professional Courses

Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.

Professional Courses
Games

Interactive Games

Fun, engaging games to boost memory, math fluency, typing speed, and English skillsβ€”perfect for learners of all ages.

games
Typing
Memory
Math
English Adventures
Knowledge

19.X.3 - Derivation of Poisson Distribution as a Limit of Binomial Distribution

Practice

Interactive Audio Lesson

Listen to a student-teacher conversation explaining the topic in a relatable way.

Introduction to the Binomial Distribution

Unlock Audio Lesson

Signup and Enroll to the course for listening the Audio Lesson

0:00
Teacher
Teacher

Alright everyone, let's start with the Binomial distribution. Can anyone tell me what it represents?

Student 1
Student 1

It models the number of successes in a fixed number of trials, right?

Teacher
Teacher

Correct! It is defined by two parameters: the number of trials, 𝑛, and the probability of success, 𝑝. Now, how do we calculate the probability of getting exactly π‘˜ successes?

Student 2
Student 2

I think it's using the formula: 𝑃(X=k) = (𝑛 choose k) * 𝑝^π‘˜ * (1βˆ’π‘)^(π‘›βˆ’π‘˜).

Teacher
Teacher

Exactly! Now, let's explore how we can derive the Poisson distribution from this. Keep in mind that we’ll discuss what happens to 𝑛 and 𝑝 as we move forward.

Conditions for Derivation

Unlock Audio Lesson

Signup and Enroll to the course for listening the Audio Lesson

0:00
Teacher
Teacher

To derive the Poisson distribution, we need to consider specific conditions. What do you think happens as the number of trials 𝑛 approaches infinity?

Student 3
Student 3

I think the individual probability 𝑝 has to go to zero.

Teacher
Teacher

Absolutely! So as we let 𝑛 go to infinity and 𝑝 go to zero, what should the product 𝑛𝑝 equal to?

Student 4
Student 4

It needs to remain constant, and we define it as πœ†.

Teacher
Teacher

Correct! This is the key to transitioning to the Poisson distribution. Now let's see what the math tells us next.

Mathematical Derivation

Unlock Audio Lesson

Signup and Enroll to the course for listening the Audio Lesson

0:00
Teacher
Teacher

Let’s take a closer look at the Binomial probability and take the limit. We have 𝑃(X=k) = (𝑛 choose k) * 𝑝^k * (1βˆ’π‘)^(π‘›βˆ’k). Can anyone help simplify this as we take limits?

Student 1
Student 1

As 𝑛 approaches infinity, the binomial coefficient becomes a bit more complicated, right?

Teacher
Teacher

Yes! But remember, as 𝑝 approaches zero, what does (1βˆ’π‘)^(π‘›βˆ’k) tend toward?

Student 2
Student 2

(1βˆ’π‘) approximates e^(-𝑛𝑝).

Teacher
Teacher

Great job! So now applying all of this, we ultimately derive the form of the Poisson probability mass function: 𝑃(X=k) = e^(-πœ†) * πœ†^k / k!. Let’s summarize what this means in real-world applications.

Applications and Connection

Unlock Audio Lesson

Signup and Enroll to the course for listening the Audio Lesson

0:00
Teacher
Teacher

Now that we've derived the Poisson distribution, let’s connect it to real-world situations. Students, can you think of any scenarios where this distribution might be applied?

Student 3
Student 3

How about modeling the number of emails received in an hour?

Student 4
Student 4

Or the count of call arrivals at a call center?

Teacher
Teacher

Exactly! The Poisson distribution is used in various fields such as telecommunications, engineering, and quality control. It helps us model events that occur independently and at a constant rate. Let’s finalize with a recap of what we have covered today.

Student 1
Student 1

We learned how to derive the Poisson distribution from the Binomial distribution and its real-world applications!

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section explains the derivation of the Poisson distribution from the Binomial distribution as a limiting case, highlighting the conditions under which this transformation occurs.

Standard

The Poisson distribution can be viewed as a limit of the Binomial distribution when the number of trials approaches infinity, the probability of success approaches zero, yet their product remains constant. This section delves into the mathematical derivation and implications of this relationship, illustrating how it connects discrete probability with continuous interpretations in practical scenarios.

Detailed

Derivation of Poisson Distribution as a Limit of Binomial Distribution

In this section, we delve into how the Poisson distribution arises as a limiting case from the Binomial distribution. The Binomial distribution models the number of successes in a fixed number of trials, represented as:
𝑃(𝑋 = π‘˜) = (𝑛 choose k) * 𝑝^π‘˜ * (1βˆ’π‘)^(π‘›βˆ’π‘˜) where:
- 𝑛 = number of trials
- 𝑝 = probability of success
- π‘˜ = number of successes

To observe the Poisson distribution, certain conditions must hold:
- The number of trials (𝑛) approaches infinity.
- The probability of success (𝑝) approaches zero.
- The product 𝑛𝑝 = πœ† remains constant.

Under these conditions, we transform the Binomial probability function into the Poisson probability mass function, yielding:
𝑃(𝑋 = π‘˜) = e^(-πœ†) * πœ†^π‘˜ / π‘˜! for π‘˜ = 0, 1, 2, ... This captures how we can model events with a known average rate (πœ†) across larger intervals or counts, especially in real-world data scenarios. The significance of this derivation extends into engineering, physics, and data modeling, reinforcing the connection between stochastic processes and deterministic equations.

Youtube Videos

partial differential equation lec no 17mp4
partial differential equation lec no 17mp4

Audio Book

Dive deep into the subject with an immersive audiobook experience.

Introduction to the Derivation

Unlock Audio Book

Signup and Enroll to the course for listening the Audio Book

The Poisson distribution is derived as a limiting case of the Binomial distribution when:
β€’ Number of trials 𝑛 β†’βˆž
β€’ Probability of success 𝑝 β†’ 0
β€’ 𝑛𝑝 = πœ† remains constant

Detailed Explanation

This chunk introduces the scenario in which the Poisson distribution can be derived from the Binomial distribution. This happens when the number of trials (n) is very large, while the probability of success (p) for each trial is very small. Despite having a high number of trials, the product of these two parameters, n times p (np), remains constant and equals πœ†, which is the average rate of events.

Examples & Analogies

Imagine a rare event like receiving a specific type of email. If a company sends out millions of emails but the chance of you receiving a particular one is very low, the situation reflects many trials (n) and a low probability of success (p). Yet, the average number of such emails (πœ†) can still be calculated.

The Binomial Probability Mass Function

Unlock Audio Book

Signup and Enroll to the course for listening the Audio Book

Let 𝑋 ∼ Binomial(𝑛,𝑝), then:
𝑛
𝑃(𝑋 = π‘˜) = ( )π‘π‘˜(1βˆ’ 𝑝)π‘›βˆ’π‘˜
π‘˜

Detailed Explanation

In this chunk, we see how the Binomial distribution creates a formula to calculate the probability of observing k successes in n trials. The formula involves a combination of n choose k, raised to the power of p for k successes and (1 - p) for the remaining failures. It's the fundamental formula for understanding how likely a certain number of occurrences is in a fixed number of trials.

Examples & Analogies

Think of lottery tickets. If you buy a certain number of tickets (n), p represents the chance of winning with each ticket. The formula helps to calculate how likely it is to win a specific number of times (k) given the total number of tickets purchased.

Taking the Limit

Unlock Audio Book

Signup and Enroll to the course for listening the Audio Book

Taking the limit as 𝑛 β†’ ∞, 𝑝 β†’ 0, such that 𝑛𝑝 = πœ†, we get:
𝑛 π‘’βˆ’πœ†πœ†π‘˜
lim ( )π‘π‘˜(1βˆ’ 𝑝)π‘›βˆ’π‘˜ =
π‘›β†’βˆž π‘˜ π‘˜!

Detailed Explanation

This chunk describes the important step of taking limits in the derivation process. By letting n approach infinity and p approach zero, while keeping their product np constant and equal to πœ†, we can derive the Poisson probability function. This shows the exact form of the Poisson distribution emerging from the formulation of the Binomial distribution under these specific conditions.

Examples & Analogies

Picture an infinitely large crowd at a concert. Each person has a tiny chance of throwing a paper plane. Even if the chance (p) is small, if there are enough people (n), the average number of planes thrown (πœ†) can still be significant. The limit process allows us to analyze at what rate this happens.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Derivation from Binomial: The Poisson distribution emerges from the Binomial distribution under certain limits.

  • Conditions: The conditions for deriving the Poisson distribution include letting the number of trials approach infinity, the probability of success approach zero, while their product remains constant.

  • Real-World Applications: The Poisson distribution has significant applications in fields like telecommunications, traffic flow, and quality control.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Example of email arrivals per hour can be modeled using the Poisson distribution to predict probabilities of receiving a specified number of emails.

  • In a manufacturing context, the Poisson distribution can model the average number of defects occurring in a given length of material.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎡 Rhymes Time

  • When events bloom, as p goes low, n's boundless trial makes πœ† grow.

πŸ“– Fascinating Stories

  • Imagine a factory where machines run and mistakes appear occasionally, but as work ramps up, odd failures become predictable, each hour pointing to a steady πœ† of errors.

🧠 Other Memory Gems

  • For Poisson: 'Many Emails All Coming (MEAC) in a steady stream' to remember that events occur independently and at a constant rate.

🎯 Super Acronyms

LIM (Limit, Infinite Trials, Mean constant) to remember the conditions for deriving Poisson from Binomial.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Binomial Distribution

    Definition:

    A discrete probability distribution that models the number of successes in a fixed number of trials.

  • Term: Poisson Distribution

    Definition:

    A discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space.

  • Term: Limiting Case

    Definition:

    A situation in calculus where a function behaves in a certain way as variables approach specific values.

  • Term: Probability Mass Function (PMF)

    Definition:

    A function that gives the probability of a discrete random variable taking a specific value.

  • Term: Ξ» (Lambda)

    Definition:

    The constant parameter of the Poisson distribution, representing the average number of events in an interval.