2 Probability Review

Today’s Learning Goals

  • Know the properties of expected value and variance of a random variable.
  • Derive mathematical properties of covariance and correlation using properties of expected value and variance.




Probability Warm Up

For a discrete random variable \(X\),

How do you calculate the expected value? (definition)

List at least two properties of the expected value.

How do you calculate the variance? (definition)

List at least two properties of the variance.

Group Activity: Covariance Challenge!

You are going to prove three REALLY IMPORTANT properties of Covariance! We’ll need these properties going forward.

Setup

  1. Introduce yourselves and check in with each other as humans.

  2. Discuss your chosen roles in facilitating your collaboration. Here are some ideas to consider:

  • One person can start with the marker
  • Another person could have course resources (Chp 2 in the Notes) available for reference and reach out to the instructor and communicate what the group is stuck on.
  • Take turns adding to the proof, explaining each step.
  • Be open and honest about how comfortable you feel about the challenge; support each other in the struggle.
  • If you feel comfortable with the problem, don’t just do it yourself. Talk through strategies/approaches that might be useful; be a guide for the group.

The goal is that everyone should feel comfortable explaining each step of the proofs.

Challenges

Using the definition of covariance of two random variables, \(Cov(X,Y) = E((X - \mu_x)(Y - \mu_y))\) where \(E(X) = \mu_x\) and \(E(Y) = \mu_y\), prove the following are true.

You may assume the properties of expected value (no need to prove those here). You may also assume #1 is true to prove #2 and assume #1 and #2 to prove #3.

  1. Prove/show: \(Cov(aX,bY) = ab Cov(X,Y)\) for random variables \(X\) and \(Y\) and constants \(a,b\). Hint: Start with \(Cov(aX,bY)\) and using properties and the definition, rewrite it as \(ab Cov(X,Y)\).

  2. Prove/show: \(Cov(X+Y,Z) = Cov(X,Z)+Cov(Y,Z)\) for random variables \(X\), \(Y\), and \(Z\). Hint: Start with \(Cov(X+Y,Z)\) and using properties and the definition, rewrite it.

  3. Prove/show: \(Cov(aX+bY,cZ + dW) = acCov(X,Z)+adCov(X,W)+bcCov(Y,Z)+bdCov(Y,W)\) for random variables \(X\), \(Y\), \(Z\), and \(W\). Hint: Start by letting \(V = cZ + dW\).