Lecture 20: Definitions and Inequalities

Course Info

Instructor

Prof. Gilbert Strang

Departments

Mathematics

As Taught In

Spring 2018

Level

Undergraduate

Topics

Learning Resource Types

Lecture Videos

Problem Sets

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Video Lectures

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

Problems for Lecture 20
From textbook Section V.1

10. Computer experiment: Find the average $A_{1000000}$ of a million random 0-1 samples! What is your value of the standardized variable $X = (A_{N} - \frac{1}{2}) / 2 \sqrt{N}$ ?

12. For any function $f (x)$ the expected value is $E [f] = \sum p_{i} f (x_{i})$ or $\int p (x) f (x) d x$ (discrete or continuous probability). The function can be $x$ or $(x - m)^{2}$ or $x^{2}$ .

If the mean is $E [x] = m$ and the variance is $E [(x - m)^{2}] = σ^{2}$ what is E $[x^{2}]$ ?

From textbook Section V.3

3. A fair coin flip has outcomes $X = 0$ and $X = 1$ with probabilities $\frac{1}{2}$ and $\frac{1}{2}$ . What is the probability that $X \geq 2 \overset{―}{X}$ ? Show that Markov’s inequality gives the exact probability $\overset{―}{X} / 2$ in this case.