Expected values of random variables

Learning Objectives Define a discrete random variable and its probability distribution. Calculate the expected value of a discrete random variable using the summation formula. Interpret the expected value as the long-run average outcome of a random experiment. Apply the concept of expected value to analyze games of chance and other real-world scenarios. Determine if a game is fair, favorable, or unfavorable by calculating the expected value of the net winnings. Use the properties of expected value to solve more complex problems. Ever wonder if that lottery ticket is *really* worth buying, or if a carnival game is designed for you to lose? 🎟️ Let's use math to find out! This tutorial introduces the concept of expected value, a fundamental tool in probability. You will l...

TermDefinitionExample Random VariableA variable, typically denoted by a capital letter like X, whose value is a numerical outcome of a random phenomenon.If you flip a coin twice, the random variable X could be the number of heads that appear. The possible values for X are 0, 1, or 2. Discrete Random VariableA random variable that can only take on a finite or countably infinite number of distinct values.The number rolled on a standard six-sided die is a discrete random variable, as its only possible values are {1, 2, 3, 4, 5, 6}. Probability DistributionA table, formula, or graph that describes the probability for each possible value of a random variable.For a fair six-sided die, the probability distribution is P(X=1)=1/6, P(X=2)=1/6, ..., P(X=6)=1/6. The sum of all probabilities must equa...

Expected Value of a Discrete Random Variable E[X] = \sum_{i=1}^{n} x_i P(X=x_i) This is the fundamental formula for expected value. To calculate it, you multiply each possible value of the random variable (x_i) by its corresponding probability P(X=x_i), and then sum all of these products together. Linearity of Expectation: Scaling and Shifting E[aX + b] = aE[X] + b If you have a random variable X, and you create a new variable by multiplying X by a constant 'a' and adding a constant 'b', the new expected value can be found by scaling the original expected value by 'a' and adding 'b'. This is very useful for simplifying calculations. Linearity of Expectation: Sum of Variables E[X + Y] = E[X] + E[Y] The expected value of the sum...