Lecture 5
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:
Sequence of n identical trials.
Two possible outcomes: success and failure.
Probability of success (p) remains constant.
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.