GE 331-Lecture 18

GE 331-Lecture 18 - Limit Theorems Last lecture, we saw...

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IE 300/GE 331 Lecture 18 Negar Kiyavash, UIUC 1 Limit Theorems • Last lecture, we saw descriptive statistics such as sample mean and variance. • Earlier in the course we had seen expected value (mean) and variance of random variables. • Intuitively we feel they are related, but can we say something precise? • For instance can we say the sample mean converges to the true mean if we have sufficient samples? • Yes, this is what limit theorems are about! + + n n X ... X n 1 μ
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IE 300/GE 331 Lecture 18 Negar Kiyavash, UIUC 2 • Markov and Chebyshev are two of the most famous, yet basic inequalities in probability theory. • Andrei Markov: (Russian mathematician) (1856 – 1922) He was a student of Chebyshev! Part of his ( known as Markov chains ) is chapter 7 of our text book. Inequalities:
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IE 300/GE 331 Lecture 18 Negar Kiyavash, UIUC 3 Markov Inequality: • Markov inequality: Consider random variable X which only takes non- negative values, then for any a>0: • Example: X~U[0,4]: Using the inequality: P(X>=3)=2/3=.67 If we look the exact probability: P(X>=3)=0.25 • Not necessarily tight (remember it is a bound not an approximation!) . ] [ ) ( P a X E a X
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Negar Kiyavash, UIUC 4 Proof: • Fix a positive number a and define r.v. Y: •N o t e t h a t Y X (Recall X was non-negative) •T h u s , E [ Y ] E[X] (*) • E[Y]=0P(Y=0)+aP(Y=a)= aP(X>=a) (**) • From (*) and (**): { 0 a X if a X if a Y < = . ] [ ) ( P
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GE 331-Lecture 18 - Limit Theorems Last lecture, we saw...

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