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Unformatted text preview: Administration Midterm regrades due by November 28th after lecture. 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 = and σ = 1 . 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 = and σ = 1 .- 2 2 . 2 . 4 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 = and σ = 1 .- 2 2 . 2 . 4 Also is “Gaussian distribution.” 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.” 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 . 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 . 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 n = A n- μ σ / √ n . 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 n = A n- μ σ / √ n . E ( A n ) 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 n = A n- μ σ / √ n . E ( A n ) = 1 σ / √ n ( E ( A n )- μ ) 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 n = A n- μ σ / √ n . E ( A n ) = 1 σ / √ n ( E ( A n )- μ ) = . Central limit theorem....
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