Discrete Random Variables Summary

Probability · Axiom Academy

SUMMARY Discrete Random Variables Let's review the key concepts of PMF, CDF, expectation, variance, and joint distributions. Definition: The PMF gives the probability that a discrete random variable X equals a specific value x Key Properties: All probabilities are non-negative ( ) and sum to 1 ( ) Example: For a fair die, for k = 1, 2, ..., 6 Why It Matters: The PMF completely characterizes the probability distribution of a discrete random variable Cumulative Distribution Function Definition: The CDF gives the probability that X is less than or equal to x Formula: is the sum of PMF values up to x Properties: Non-decreasing, right-continuous, ranges from 0 to 1 Why It Matters: Useful for computing probabilities over intervals: Example Recap: Computing E[X] and Var(X) Step 1 - Identify the PMF: Determine for all possible values of X Step 2 - Calculate E[X]: Compute by multiplying each value by its probability and summing Step 3 - Calculate E[X²]: Compute by multiplying each squared value by its probability and summing Step 4 - Find Variance: Use the formula to get the variance, and take the square root for standard deviation Sum Property: (holds for any X and Y) Key Insight: Linearity holds regardless of whether X and Y are independent Constant: (constants have no variability) Scaling: (variance scales quadratically) Translation: (shifting doesn't change spread) Independence: If X and Y are independent, Joint Distributions & Relationships

This is the written version of the interactive lesson above. See the full Probability course.