Lecture Details

• Instructor: Samuel Wylde

• Course: Econ 270: Statistics for Economics

• Department: Economics, University of Illinois Chicago

• Semester: Fall 2024

Chapter 5: Discrete Probability Distributions

• Sections:

  • 5.1 - Random Variables

  • 5.2 - Developing Discrete Probability Distributions

  • 5.3 - Expected Value and Variance

  • 5.4 - Bivariate Distributions and Covariance

  • 5.5 - Binomial Probability Distribution

  • 5.6 - Poisson Probability Distribution

5.1 Random Variables

• Definition: A random variable is a numerical description of the outcome of an experiment.

• Types:

  • Discrete Random Variable: Can assume a finite number of values or an infinite sequence of values.

  • Continuous Random Variable: Can assume any numerical value in an interval or collection of intervals.

• Clarification: Random in terms of the outcome of an experiment, not randomness in the sense of being arbitrary.

• Examples:

  • Household size: Discrete

  • Distance from home to work: Continuous

  • Age: Discrete

  • Driving speed: Continuous

5.2 Discrete Probability Distributions

• Probability Distribution: Describes how probabilities are distributed over the values of a random variable.

• Form Representation: Can be illustrated via tables, graphs, or formulas.

• Types:

  • First Type: Assigning probabilities based on experimental outcomes.

  • Second Type: Using a mathematical formula to compute probabilities for each value.

• Notation:

  • Random variable X (e.g., number of 3-point shots made in a game).

  • Specific value of X is denoted as x (e.g., 12 shots).

  • Probability function f(x) = P(X = x).

• Conditions for Discrete Probability Function:

  • f(x) ∈ [0, 1]

  • Σ f(x) = 1

• Methods for Assigning Probabilities:

  • Classical, subjective, and relative frequency methods.

Example: JSL Appliances

• Tabular representation of TV sales:

  		Units Sold Number of Days f(x)
  0	80	0.4
  1	50	0.25
  2	40	0.2
  3	10	0.05
  4	20	0.1

5.3 Expected Value and Variance

• Expected Value: Measure of central location. Defined as a weighted average.

  • = µ = Σ xf(x)

• Variance: Describes variability in values, calculated as a weighted average of squared deviations.

  • = σ² = Σ(x − µ)²f(x)

• Standard Deviation: σ = √Var[X]

5.4 Bivariate Distributions

• Definition: Involves two random variables.

  • Example: Daily car sales at two dealerships (Geneva and Saratoga).

• Covariance: Measures association between two random variables.

  • Formula: Cov(X,Y) = σXY = [Var(X + Y) - Var(X) - Var(Y)]/2

  • Positive covariance indicates a positive relationship.

5.5 Binomial Probability Distribution

• Characteristics of a Binomial Experiment:

  1. Sequence of n identical trials.

  2. Two possible outcomes: success and failure.

  3. Probability of success (p) remains constant.

  4. Trials are independent.

• Binomial Probability Function: f(x) = n! / [x!(n-x)!] p^x (1-p)^(n-x)

  • Where:

    • x = number of successes

    • n = number of trials

    • p = probability of success

Example: Evans Electronics

• Question: Probability that 1 out of 3 randomly chosen employees will leave.

• Calculate: By applying the binomial function with n = 3, x = 1, and p = 0.1.

  • Resulting in f(x) = 0.243.

5.6 Poisson Probability Distribution

5.6 Poisson Probability Distribution

• Definition: A Poisson distribution models the number of times an event occurs in a fixed interval of time or space.

• Characteristics:

  • Suitable for rare events.

  • Events occur independently.

  • The average rate (λ) is constant.

• Poisson Probability Function:

  • f(x) = (e^(-λ) * λ^x) / x!

  • Where:

    • x = actual number of events

    • λ = expected number of events in the interval

• Example:

  • If the average number of cars arriving at a service station in an hour is 3, then λ = 3. The probability of exactly 2 cars arriving in an hour can be calculated using the Poisson function.