Gaussian rv with mean and variance w n 2 its pdf

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Unformatted text preview: a) , x ≥ a, 0, otherwise. ENGG2430C lecture 8 16 / 26 Gaussian (Normal) Random Variable Consider a Binomial r.v. Y = ∑n=1 Xi where Xi are i.i.d Bernoulli r.v.s i with parameter p . E [Y ] = np , Var(Y ) = np (1 − p ). 1 1 1 Consider n Y = n ∑n=1 Xi . Keep p fixed, let n → ∞. n ∑n=1 Xi → W , a i i 2. Gaussian r.v. with mean µ and variance σ W ∼ N (µ , σ 2 ). Its PDF is given by (w − µ )2 1 e − 2σ 2 , −∞ < w < ∞. fW (w ) = √ 2πσ 2 M. Chen (IE@CUHK) ENGG2430C lecture 8 17 / 26 Gaussian (Normal) Random Variable (From wikipedia) FACTS: E [W ] = µ , Var(W ) = σ 2 Let S = (W − µ )/σ , then S ∼ N (0, 1) No close form for CDF, but there are table to check (for standard Gaussian) Example: W ∼ N (0.25, 0.00375), −0. 0. P (W ≤ 0.1) = P W0.0625 ≤ 0.1−0625 = CDF (−2.5) ≈ 0.006. 0. M. Chen (IE@CUHK) ENGG2430C lecture 8 18 / 26 Summary of Concepts Probability model Ω P (A) Discrete r.v. r.v. X , Y pX ( x ) Continuous r.v. r.v. X , Y fX (x ) FX (x ) E [X ], var(X ) P (A ∩ B ) P (A|B ) M. Chen (IE@CUHK) pX ,Y (x , y ) pX |Y (x |y ) ENGG2430C lecture 8 fX ,Y (x , y ) fX |Y (x |y ) 19 / 26 Joint PDF and Independence Let X and Y be two continuous random variables, and their joint PDF be fX ,Y (x , y ) ˆ P (B ) = fX ,Y (x , y ) dx dy B P (x ≤ X ≤ x + δ , y ≤ Y ≤ y + δ ) ≈ fX ,Y (x , y )δ 2 ˆ ∞ˆ ∞ E [g (X , Y )] = g (x , y )fX ,Y (x , y ) dx dy −∞ −∞ ˆ∞ fX ,Y (x , y ) dy · δ f X (x )δ ≈ P (x ≤ X ≤ x + δ ) ≈ −∞ fX ,Y (x , y ) fX |Y (x |y ) = fY (y ) X and Y are independent if fX ,Y (x , y ) = fX (x )fY (y ) or fX |Y (x |y ) = fX (x ) for all x , y M. Chen (IE@CUHK) ENGG2430C lecture 8 20 / 26 Laws in Terms of PDF Bayes’ rule pX ( x ) pY (y ) fX (x ) fX |Y (...
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This document was uploaded on 03/31/2014.

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