Probability and Random Variable

Chapter 11: Defining Probability

• Tree Diagrams: Visual representation of possible outcomes in a procedure.

  • Useful for small sample spaces.

• Example: Virus detection test probabilities.

  • Probability infected: 𝑃(𝐼) = 0.005.

  • Probability negative test: 𝑃(−) = 0.94635.

Chapter 13: Random Variables

• Random Variable: X, a numerical value from a procedure determined by chance.

• Types:

  • Discrete: Finite/countable values (e.g., coin toss).

  • Continuous: Infinitely many values (e.g., height).

• Probability Distribution: Table of outcomes and their probabilities.

  • Example: Rolling a die, sum = 1/2 for outcomes > 3.

• Distributions: Variance and Standard Deviation calculations.

  • Expected value: 𝐸(𝑋) = SUM(x ⋅ P(x)).

Bookstore Revenue Model

• Revenue per student: 0% purchase, 55% buy textbook, 25% buy both.

• Expected revenue: $117.85, variance: 3659.

• Expected revenue from 40 students: $4714.

Life Insurance Example

• Expected profit from policy: $316 loss per person.

  • Expected benefits: 185 (if survives) vs. -100,000 (if not).

Linear Combinations of Random Variables

• Formulas:

  • Expected Value: 𝐸(aX + bY) = a𝐸(X) + b𝐸(Y).

  • Variance: 𝜎^2(X+Y) = 𝜎^2(X) + 𝜎^2(Y).

Tests and Exercises

• Blood test: probabilities involving Parkinson’s detection.

• Home-to-school commute scenarios (routes) leading to respective lateness probabilities.

Random Variables Exam Questions

• Examples to determine random variables: aspects of discrete vs continuous.

• Construction of distribution tables for various scenarios (e.g., child gender).

• Calculation of probabilities within a family of births and rolling dice outputs.

You would use the expected value formula, E(X) = \Sigma(x \cdot P(x)), when you want to calculate the average outcome of a discrete random variable. This is useful in scenarios like determining the average sum when rolling a die, calculating the expected revenue per student in a bookstore, or determining the expected profit from a life insurance policy per person. It gives you a single value that represents the long-term average of the outcomes.

The formulas for linear combinations of random variables, E(aX + bY) = aE(X) + bE(Y) and \sigma^2(X+Y) = \sigma^2(X) + \sigma^2(Y), are used when you are dealing with the sum or linear combination of two or more independent random variables (X and Y). The expected value formula for linear combinations helps you find the expected value of a combined outcome, where individual random variables might be scaled. The variance formula for linear combinations helps you understand the total variability or risk when independent random outcomes are added together, common in situations involving multiple uncertain factors like combining revenues from different product lines or assessing overall risk from various investments.