Econ 120A - Discrete Probability Distributions

Housekeeping

• Exam 1 on Monday (4/28)

  • Covers materials on Descriptive Statistics and Probability Theory.

  • Probability Theory will be finished today.

• Exam 1 Review Session over Zoom on Sunday (4/27) at 10 am.

  • TA will review MC & FB Practice Questions.

• Problem Sets 1 & 2 are due by 11:59 pm on Sunday (4/27).

• Check TA session recordings.

• Exam 1 practice questions with solutions and a formula sheet are available on Canvas.

Exam Seating

• Check the assignment's comment section for your seat assignment:

  • Click on Grades in the course menu.

  • Click on the comments icon for "Midterm 1 Seat Assignment."

  • Note your seat assignment in the comments. For example: "LEDDN AUD, seat J-4"

Discrete Probability Distributions

• Topics covered:

  • Discrete Random Variables

  • Probability Distribution

  • Mean of a Random Variable

  • Variance of a Random Variable

  • Expected Values

  • Binomial Distribution

  • Characteristics of Binomial RVs

  • Mean and Variance of Binomial RVs

Random Variable

• Sometimes, the possible outcomes of an experiment are not numerical values.

  • Example: Tossing a fair coin.

  • Two possible outcomes: H, T.

  • S = {H, T}

  • Outcomes are not numbers, making algebraic computations impossible.

• To enable algebraic computations on experiment outcomes, translate them to real numbers.

Random Variable - Formal Definition

• A random variable is a mathematical function that assigns a real number to each outcome in the sample space of an experiment.

• It's a "rule" assigning a real number (x) to each outcome in an experiment's sample space.

Random Variable - Coin Toss Example

• Experiment: Tossing a fair coin.

• Possible outcomes: H, T.

• S = {H, T}

• Random variable X:

  • X(T) = 0, X(H) = 1

  • X takes the value 0 for T (tails) and 1 for H (heads).

• Capital letters denote random variables, and lowercase letters denote the possible values the variable can take.

• X is the complete rule assigning numbers to coin flip outcomes.

• x represents possible values of X: in this example, x = 0 or x = 1.

Another Example

• Throw of a die:

  • X = numbers of dots shown

  • X = 1 if an odd number of dots, X = 0 if even

  • X = 1 if 3 or fewer dots, X = −1 if 4 or more

Random Variable Types

• Discrete: Possible values can be listed.

• Continuous: Possible values cannot be listed (too many to list).

• Discrete random variable:

  • May assume a finite or an infinite sequence of values (e.g., 0, 1, 2, …).

• Continuous random variable:

  • May assume any numerical value in an interval or collection of intervals.

Discrete Random Variable with a Finite Number of Values

• Example: JSL Appliances

  • X = number of TVs sold at the store in one day.

  • X can take on 5 values (0, 1, 2, 3, 4).

  • The number of TVs sold can be counted, with a finite upper limit (number of TVs in stock).

Discrete Random Variable with an Infinite Number of Values

• Example: JSL Appliances

  • X = number of customers arriving in one day.

  • X can take on the values 0, 1, 2, . . .

  • The number of customers arriving can be counted, but there is no finite upper limit.

Random Variables - Illustration

• Examples:

  • Family size: X = Number of dependents reported on tax return (Discrete).

  • Distance from home to stores on a highway: X = Distance in miles from home to the store site (Continuous).

  • Own dog or cat: x = 1 if own no pet; = 2 if own dog(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s) (Discrete).

Probability Distribution of a Discrete Random Variable

• A probability distribution is an assignment of probability to each possible value of the random variable.

• For a discrete random variable, it's a list of probabilities, one for each possible value of the RV.

• The probability of each value represents the chance that the value will occur.

• Probability can be expressed using a graph or a formula (equivalent representations).

• Notation: P(X=x) = p(x) = probability that the random variable X takes on the specific value x.

Example 1

• Rolling a die:

  • X = value showing up on the die face

  • X has six possible values: 1, 2, 3, 4, 5, and 6.

  • Each value is equally likely.

Example 2

• Tossing a coin three times.

  • Sample space: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

  • X = total number of heads in the three flips.

  • The random variable X has 4 possible values.

• P(X < 2) = p(0) + p(1) = 1/8 + 3/8 = 4/8

Example 3

• X = Sum of two dice

Example 4

• Example: JSL Appliances

• Using past data on TV sales, a tabular representation of the probability distribution for sales was developed.

Example 5

• The discrete uniform probability distribution is the simplest example of a discrete probability distribution given by a formula.

• The discrete uniform probability function is given by:
  p(x) = 1/n

• where:

  • n = the number of values the random variable may assume

• The values of the random variable are equally likely.

Basic Properties of Probability Distributions

• For any random variable X:

  • 0 ≤ p(x) ≤ 1

  • \sigma p(x) = 1

Mean of a Discrete Random Variable

• The mean of a discrete random variable X is given by
  \mu = \sigma xp(x)

• The mean of X is a weighted average of all possible values of X, in which the weights are the probabilities that those values occur.

Variance of a Discrete Random Variable

• The variance of a discrete random variable X is given by
  \sigma^2 = \sigma (x - \mu)^2 p(x)

• The variance of X is a weighted average of squared deviations from the mean, in which the weights are the probabilities that those values occur.

To Sum Up

• Population Mean
  \mu = \sigma xp(x)

• Population Variance
  \sigma^2 = \sigma (x - \mu)^2 p(x)