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lec-33.handout

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CS70: Satish Rao: Lecture 33. Continuous Probability 1. Normal (Gaussian) Distribution. 2. Joint distributions. 3. Buffon’s needle. 4. Begin inference.
Normal Distribution. For any μ and σ , a normal random variable has pdf f ( X ) = 1 2 πσ 2 e - ( x - μ ) 2 / 2 σ 2 . Standard normal has μ = 0 and σ = 1 . - 2 0 2 0 0 . 2 0 . 4 Also is “Gaussian distribution.”

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Central limit theorem. Law of Large Numbers: For any set of independent identically distributed random variables, X i , A n = 1 n X i “tends to the mean.” Say X i have expecation μ = E ( X i ) and variance σ 2 . Mean of A n is μ , and variance is σ 2 n . Let A 0 n = A n - μ σ / n . E ( A 0 n ) = 1 σ / n ( E ( A n ) - μ ) = 0 . Var ( A 0 n ) = 1 σ 2 / n Var ( A n ) = 1 . Central limit theorem: As n goes to infinity the distribution of A 0 n approaches the standard normal distribution. Pr [ A 0 n α ] 1 2 π Z α e - x 2 / 2 dx .
Show pictures here...of binomial converging to Gaussian.

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Standard Normal.
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