Introduction to Random Variables and Expected Values

A random variable is any numerical outcome where the exact result is uncertain.
It allows for making predictions about future events rather than merely describing past or present states.
Examples:
- The number of passengers who don't show up for a flight.
- The number of points scored in a sports championship.
- The number of students attending a class (e.g., out of 65 enrolled, less than 65 are expected for any given session due to illness or other reasons).
Even though individual outcomes are uncertain, statistical insights and probability models can be applied.

The probability distribution function (or probability mass function for discrete variables) defines the probability of each possible outcome of a random variable.
Example: Hyde Park Household Size
- Scenario: A student needs moving help in Hyde Park and knocks on a neighbor's door.
- The random variable is the number of people living in the house next door.
- US census data provides probabilities for different household sizes:
  - P(X=1) = 28.0\% (one person)
  - P(X=2) = 33.6\% (two people)
  - P(X=3) = 15.5\% (three people)
  - (Implicitly, probabilities for 4, 5, ext{etc.} people also exist).
- This distribution helps estimate the likelihood of getting help.
Visualization: Similar to a histogram, but the height of each bar represents the probability of an outcome, not just its count.
Notation:
- Big X: Represents the random variable itself (e.g., number of people in a household).
- Little x: Represents a specific value the random variable X can take (e.g., 1, 2, 3, ext{etc.} people).

Definition: Outcomes can be listed or counted; there's a finite or countably infinite number of possibilities.
Key Criterion: Can you make a list of all possible outcomes?
Common Characteristic: Often (but not exclusively) whole numbers.
Examples:
- Number of days in a year with recorded precipitation (0 to 365 options).
- Count of tornado warnings (a fixed, finite number).
- Room numbers (e.g., UTC room numbers are decimals, but finite and listable).
- Shoe sizes (e.g., whole numbers and halves, but not arbitrary decimals).
Also referred to as a Probability Mass Function (PMF).

Definition: Outcomes can take any value within a given range; there are infinitely many possibilities, usually involving decimal places.
Key Characteristic: Can be refined indefinitely by adding more decimal places.
Examples:
- Speed of a tennis ball (e.g., 70 mph, 70.1 mph, 70.15 mph, etc.).
- Annual rain accumulation in inches (e.g., 120.1 inches, 120.15 inches, etc.).

The expected value of a random variable X is its long-run average value.
It's a weighted average, where each possible outcome is weighted by its probability of occurrence.
Contrast with Simple Average: A simple average (arithmetic mean) assumes all outcomes are equally likely, which is rarely true in real-world probability distributions (e.g., household sizes). The expected value accounts for varying probabilities.
Formula for Discrete Random Variables:
E[X] = ext{Sum over all possible values of } x (x imes P(X=x))
E[X] = oldsymbol{ ext{E}}[X] = oldsymbol{ ext{x}} imes oldsymbol{ ext{P}}(oldsymbol{ ext{X}}=oldsymbol{ ext{x}})
For Continuous Variables: The concept is the same but involves calculus (integration) instead of summation.

Life and Disability Insurance Company
- Scenario: A company with 1,000 customers.
- Events and Payouts:
  - Death: Probability 1/1000 = 0.001, Payout \text{USD } 100,000
  - Disability: Probability 2/1000 = 0.002, Payout \text{USD } 50,000
  - No event: Probability 997/1000 = 0.997, Payout \text{USD } 0
- Expected Payout per Customer:
  E[ ext{Payout}] = (0.001 imes 100,000) + (0.002 imes 50,000) + (0.997 imes 0)
  E[ ext{Payout}] = 100 + 100 + 0 = ext{USD } 200
- Implication: To make a profit, the company must charge customers more than \text{USD } 200 per year in premiums.
Bookstore Sales
- Expected value of books a customer buys helps estimate sales, stock bags, and guide store decisions.
- Provides a