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
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 \(X=\left(A_N-\frac{1}{2}\right) / 2\sqrt{N}\)?
12. For any function \(f(x)\) the expected value is \(\hbox{E}[f]=\sum p_i\, f(x_i)\) or \(\int p(x)\,f(x)\,dx\) (discrete or continuous probability). The function can be \(x\) or \((x-m)^2\) or \(x^2\).
If the mean is \(\hbox{E}[x]=m\) and the variance is \(\hbox{E}[(x-m)^2]=\sigma^2\) what is E\(\boldsymbol{[x^2]}\)?
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 \(X\geq 2\overline{X}\)? Show that Markov’s inequality gives the exact probability \(\overline{X}/2\) in this case.
