Lecture 20: Definitions and Inequalities

Description

This lecture continues the focus on probability, which is critical for working with large sets of data. Topics include sample mean, expected mean, sample variance, covariance matrices, Chebyshev’s inequality, and Markov’s inequality.

Summary

\(E[x] = m =\) average outcome weighted by probabilities
\(E\) uses expected outcomes not actual sample outcomes.
\(E[(x - m)^2] = E[x^2] - m^2\) is the variance of \(x\).

Markov’s inequality Prob[\(x \geq a\)] \(\leq\) mean\(/a\) (when all \(x\)’s \(\geq\) 0)
Chebyshev’s inequality Prob[|\(x\) - mean| \(\geq\) \(a\)] \(\leq\) variance\(/a^2\)

Related sections in textbook: V.1, V.3

Instructor: Prof. Gilbert Strang