Unit 7 of 13
Joint Discrete Laws and Conditional Averages
- 8 lessons
- Videos 44 to 51
A joint distribution describes how two variables occur together. This unit covers marginal and conditional distributions, independence, covariance and correlation, and conditional expectation.
Marginal distributions come from summing out the other variable, and they lose the information about how values pair up. Conditional distributions renormalize one row or column of the joint table. Independence requires the joint law to factor at every pair of values, and zero covariance does not imply independence. The law of total expectation builds an overall mean from conditional means, the law of total variance splits variance into within-group and between-group parts, and together they give the mean and variance of a sum with an independent random number of terms.
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.
Joint, Marginal and Conditional Distributions
Joint Probability Mass Functions
How a joint probability mass function gives a probability to every pair of values of two discrete random variables, and how to check a joint table, read its support and add cells for events and functions of the pair.
Video 44
Marginal Probability Distributions
How to find the marginal distribution of each variable from a joint probability table by summing out the other variable, why the margins are ordinary PMFs, and what they lose about how values pair up.
Video 45
Conditional Probability Mass Functions
How to turn one row or column of a joint table into a conditional PMF, why the condition needs positive probability, how reversing the condition changes the denominator, and how conditional means and joint masses follow.
Video 46
Independence and Covariance
Independent Random Variables
What it means for random variables to be independent, how the product law and CDF factorization test a joint law at every cell and threshold, why separate functions stay independent, and what mutual independence adds.
Video 47
Covariance, Correlation and Sums
How covariance and correlation measure the way two random variables move together, why zero covariance does not imply independence, how covariance enters the variance of a sum, and why the size of a correlation is at most 1.
Video 48
Conditional Expectation and Total Variance
Conditional Expectation and the Tower Rule
What conditional expectation is, how the conditional mean function becomes a random variable, why the law of total expectation recovers the overall mean, and why the conditional mean minimizes expected squared error.
Video 49
The Law of Total Variance
How the law of total variance writes a variance exactly as within-group variation, the expected conditional variance, plus between-group variation, the variance of the conditional means.
Video 50
Random Sums with Independent Counts
How to find the mean and variance of a random sum, a total whose number of terms is a random count, by conditioning on a count independent of the IID terms, and why the empty sum equals zero.
Video 51