QMS 230 Lecture Notes - Binomial and Poisson Distribution

Course Overview

• Course Title: W26 QMS 230

• Lecture Number: 3

• Topics Covered: Binomial and Poisson Distribution

• Instructor: Dr. Boža Tasić

• Date: January 25, 2026

Learning Objectives for Binomial Distribution

• Understanding Basic Probability and its rules

• Basic Counting Rules

• Binomial Distribution Formula

• Expected Value and Standard Deviation of Binomial Distribution

Basic Counting Rules

Rule 1: Outcomes of Mutually Exclusive Events

• For any one of k different mutually exclusive and collectively exhaustive events occurring in n trials, the number of possible outcomes is given by:

  • $k^n$

• Example:

  • Rolling a fair die 3 times: $6^3 = 216$ possible outcomes.

  • Tossing a coin 5 times: $2^5 = 32$ possible outcomes.

Rule 2: Product of Different Events

• For k1 events on the first trial, k2 events on the second trial, …, kn events on the nth trial, the total number of possible outcomes is:

  • $k<em>1 imes k</em>2 imes … imes k_n$

• Example:

  • Selecting from 2 museums, 4 restaurants, and 6 movies:

  • Total combinations = $2 imes 4 imes 6 = 48$ different possibilities.

Rule 3: Arrangements of n Items

• The number of ways to arrange n items in order is:

  • $n! = n imes (n-1) imes (n-2) … imes 1$

• Example:

  • Arranging 6 books on a shelf:

  • Total arrangements = $6! = 720$

• For any k < n, it holds that:

  $n! = n imes (n-1) imes … imes (k+1) imes k!$

Rule 4: Selecting Objects (Combinations)

• The number of ways to select x objects from n objects, irrespective of order, is defined as:

  • $nC_x = \frac{n!}{x!(n-x)!}$

• Example:

  1. Selecting 4 out of 6 books:

  • Total combinations = $6C4 = \frac{6!}{4! imes 2!} = 15$.

  1. Selecting 4 ice cream scoops from 20 flavors:

  • Total combinations = $20C4$ is calculated as $\frac{20!}{4! imes 16!} = 4845$.

Probability Distributions

• Introduction to three types of probability distribution: Binomial and Poisson distributions which have specific mathematical models for computation.

Binomial Distribution

• Definition: A binomial distribution provides a mathematical model to compute the probability of occurrence of a particular outcome of a discrete random variable, given a defined number of observations n.

• Conditions for Binomial Distribution:

  • Fixed number of observations n

  • Each observation results in one of two possible outcomes (success or failure).

  • Example: Coin toss - head (success) or tail (failure).

  • Constant probability of success π for each trial, with the probability of failure being (1 - π).

• Random Variable: The random variable associated with a binomial experiment is represented as X = number of successes.

Binomial Probability Distribution Formula

• To calculate the probability that X takes on a specific value:

  • $P(X = x) = \frac{n!}{x!(n-x)!} \pi^x (1 - \pi)^{n-x}$

• Where:

  • n = number of trials

  • x = number of successes

  • π = probability of success in each trial

Example 1: Coin Toss Problem

• Problem: Toss a coin 6 times; find the probability of getting exactly 2 tails.

• Parameters:

  • X = number of tails;

  • n = 6;

  • π = 0.5 (getting tails).

Calculating Probability

1. Apply the binomial formula:
   $P(X = 2) = \frac{6!}{2! (6-2)!} \cdot (0.5)^2(0.5)^4$

2. Determine the combinations:

   • Number of outcomes for 2 tails:

   • $\frac{6!}{2!4!} = 15$

3. Thus, the probability:
   $P(X = 2) = 15 \cdot 0.5^6 = 0.2344$

Using Calculator to Compute P(X = x)

Step-by-step process using CASIO calculator:

1. Select: STAT F5 (DIST), F5 (BINM), F1 (Bpd).

2. Input data:

   • x: 2

   • numtrials: 6

   • p: 0.5

3. Result: Binomial P.D = approx 0.234375.

Example 2: Investment Survey

• Problem: In a condo with 18 adults aged 30-40, need probabilities concerning mutual fund investments (43% have investments).

• Parameters:

  • X = number of adults with investments;

  • n = 18;

  • π = 0.43.

Probability Calculations:

1. a) P(X = 5):

   • Formula: $\frac{18!}{5!(18-5)!} (0.43)^5(0.57)^{13} = 0.0845$.

2. b) P(X ≤ 2): Sum probabilities for P(X = 0), P(X = 1), and P(X = 2).

3. For at most 2:

   • $P(X = 0) + P(X = 1) + P(X = 2) = 0.00004 + 0.00055 + 0.00351 = 0.0041$.

4. d) P(3 ≤ X ≤ 9): can be calculated as follows:

   • Compute P(X ≤ 9) and P(X ≤ 2): resulting in + probabilities summing to 0.7955.

Binomial Template Notation

• Denote the formula for binomial probabilities as:

  • $P(X = x) = Bpd(x, n, π)$

  • Cumulative probabilities as:

  • $P(X ≤ x) = Bcd(x, n, π)$.

Poisson Distribution

Learning Objectives for Poisson Distribution

• Characteristics of a Poisson Experiment

• Poisson Distribution Function

• Expected Value and Standard Deviation of the Poisson Distribution

Poisson Probability Characteristics

• The Poisson distribution is used for count-based events within a unit of time or space.

• Situations include:

  • Number of casualties occurring in an area.

  • Number of defects in manufactured goods.

• Characteristics of a Poisson Event:

  • Successes occur randomly and independently in the given time/space.

  • Success probability remains constant.

  • Statistical expectation for occurrences is denoted by λ.

Poisson Probability Distribution Function

• Formula:

  • $P(X = x) = \frac{e^{-λ} λ^x}{x!}$

• Where:

  • x = number of events in an area of opportunity;

  • λ = expected number of events;

  • e = approx 2.71828.

Computing Poisson Probabilities

1. Example 1: Probability of 12 customers using a bank machine in an hour, where:

   • λ = 15 (average rate per hour).

2. Parameter Setup:

   • P(X = 12) = Ppd(12, 15) = 0.0829 (about 8.3% chance).

3. Short Interval: To find the chance of fewer than 3 customers in 10 minutes, recalculate λ for that time:

   • λ = 15 (customers/hour) => λ = 2.5 (customers/10 minutes).

   • Calculate: P(X < 3) = Pcd(2, 2.5) = 0.5438.

Expected Value and Standard Deviation for Poisson Distribution

• Mean (Expected Value) = λ, with formula:

  • $E(X) = λ$

• Standard Deviation = $σ = \sqrt{λ}$.

Conclusion

• A comprehensive understanding of Binomial and Poisson distributions is critical for calculating probabilities based on discrete events in defined trials or intervals. Knowledge of these distributions, along with competence in using calculation tools, is essential for effective analytics in various fields such as finance, healthcare, and manufacturing.

Graphical Representation

• The shape of the Poisson distribution varies depending on the mean number of events, λ, and is graphed variably for demonstration in lectures.

Additional Resources

• Poisson Distribution Applet for visual understanding can be found at: https://homepage.divms.uiowa.edu/~mbognar/applets/pois.html