Chapter_5_Slides_Jaggia

Chapter 5: Discrete Probability Distributions

Overview of Discrete Probability Distributions

• Key Topics:

  • Description of discrete random variables and their probability distributions

  • Summary measures for discrete random variables

  • Calculation and interpretation of probabilities for:

    • Binomial random variables

    • Poisson random variables

    • Hypergeometric random variables

5.1 Definition of Discrete Random Variables

• Random Variable: Function that assigns numerical values to outcomes of an experiment.

  • Captures uncertainty and summarizes outcomes using numerical values

  • Denoted conventionally by X.

Types of Random Variables

1. Discrete Random Variables

   • Assume a finite number of values or an infinite countable sequence.

   • Examples: Number of correct answers on an exam.

2. Continuous Random Variables

   • Can assume any numerical value in an interval.

   • Example: Average score on an exam.

Infinite Values in Discrete Random Variables

• Example: Number of cars arriving at a toll booth.

  • Let x = number of cars arriving in one day, where x = {0, 1, 2, ...}.

  • Infinite potential without finite upper limit.

Characteristics of Random Variables

• Experiment: e.g., inspect shipment of radios, operate a restaurant, fill a soft drink can, operate a bank.

• Each scenario yields possible values of the random variable:

  • Inspect radios: 0-50 defective (discrete finite).

  • Restaurant customer count: 0 to infinity (discrete infinite).

  • Can volume: 0 ≤ x ≤ 12.1 ounces (continuous).

  • Time between customer arrivals: x ≥ 0 (continuous).

Discrete Probability Distributions

• Describes how probabilities are distributed across values of a random variable.

• Can be represented in tables, graphs, or formulas.

• Types of Distribution:

  • Based on rules for assigning probabilities (empirical).

  • Uses mathematical formulas to determine probabilities.

Definition of Probability Function

• Probability distribution defined by a probability function, denoted P(X = x).

  • Gives probability that random variable X equals a specific value.

  • Conditions:

    • 0 ≤ P(X = xi) ≤ 1

    • Total probability = 1.

Methods for Assigning Probabilities

1. Classical Method

2. Subjective Method

3. Relative Frequency Method.

Example: DiCarlo Motors Sales Data

• Sample data showing number of cars sold vs. frequency:| Cars Sold | Days | P(X = x) | |-----------|------|-----------| | 0 | 54 | 0.18 | | 1 | 117 | 0.39 | | 2 | 72 | 0.24 | | 3 | 42 | 0.14 | | 4 | 12 | 0.04 | | 5 | 3 | 0.01 | | Total | 300 | 1.00 |

Graphical Representation

• Probability distribution graphically represented, showing relative likelihoods of various sales outcomes.

Probability Mass Function

• A formula that assigns probabilities to values of X in case of discrete variables.

• Common discrete distributions:

  • Discrete Uniform Distribution

  • Binomial Distribution

  • Poisson Distribution

  • Hypergeometric Distribution.

Expected Value of a Random Variable

• The expected value (mean) measures a random variable's central location.

• Calculated as:

  • E(X) = Σ xP(X = x)

• It reflects a weighted average considering probabilities.

  • It need not be an actual value that the random variable can assume.

Variance and Standard Deviation

• Variance: Measures variability and is computed as:

  • Var(X) = Σ [X - E(X)]² P(X = x)

• Standard deviation:

  • S(X) = √Var(X).

Example Calculation: DiCarlo Motors

• Expected number of cars sold: E(X) = 1.50.

• Variance: Var(X) = 1.25.

• Standard deviation: S(X) = 1.118 cars.

Binomial Probability Distribution

• Based on Bernoulli process conditions:

  • Identical trials, independence, two outcomes (success/failure), constant probability for success.

• Focus on number of successes in n trials.

  • Binomial random variable, X, can take values from 0 to n.

Binomial Probability Function

• Formula: P(X = x) = (n! / (x!(n-x)!)) p^x (1-p)^(n-x).

Examples in Context

• Example of customer purchases at a store with calculated probabilities for different outcomes and expected values versus actual observed results.

Excel Application

• Use of Excel to calculate probabilities, expected values, variance, and standard deviation.

• Functions include:

  • BINOM.DIST for specific probability calculations.

  • Formulas derived from data to summarize and analyze business outcomes.

Conclusion

• Key understandings about discrete probability distributions facilitate better decision-making in business contexts through probabilistic modeling.