Unit 4 of 13

Discrete Random Variables and Expected Value

  • 6 lessons
  • Videos 25 to 30

A random variable assigns a number to each outcome. This unit builds its probability mass function and cumulative distribution function and defines expected value.

The probability of a numerical event is the sum of the masses of the values it includes, and the CDF accumulates those masses up to every threshold, with a jump at each possible value. When several inputs give the same output of a function, their masses are combined. Expected value is the probability-weighted average of the values. It need not be a possible value, and with infinitely many values it need not be finite. The unit ends with expected payoff and expected cost in decisions, break-even probabilities and the risk that the model itself is wrong.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Random Variables and Their Distributions

  1. Discrete Random Variables

    What a random variable is, how one observed realization differs from its distribution, and when a random variable is discrete, with a finite or countably infinite support whose values need not be integers.

    Video 25

  2. Probability Mass Functions

    How a probability mass function lists the probability of each value of a discrete random variable, how to check nonnegative masses and total mass one, find a normalizing constant, and add masses to get event probabilities.

    Video 26

  3. Cumulative Distribution Functions

    How the cumulative distribution function of a discrete random variable adds up probability to each threshold, rises only by jumps equal to the masses, is right-continuous, and gives tail and interval probabilities.

    Video 27

  4. Functions of Discrete Random Variables

    How to find the distribution of a function of a discrete random variable by adding the masses in each preimage, for one-to-one maps, many-to-one maps and the positive part.

    Video 28

Expected Value

  1. Expected Value

    What the expected value of a discrete random variable is, how it differs from the mode, how to find a conditional mean and an expected net gain, and when the expectation is infinite or undefined.

    Video 29

  2. Expected Value and Decisions

    How expected value supports decisions, with gross and net payoff, fair price, break-even probability and expected cost, and why the decision criterion and model risk must be stated.

    Video 30