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Tutorial 4 Probability Density 2 ENGG2450A Tutors The Chinese University of Hong Kong 13 February 2017 1/5 The Uniform Distribution 2/5 The uniform distribution Definition Uniform distribution. We say a continuous random variable X has uniform distribution on [α, β] if it has the following probability density function, 1 β−α , for α < x < β, f (x) = 0, otherwise. Mean of the uniform distribution is given by µ= β+α . 2 Variance of the uniform distribution is given by σ2 = 1 (β − α)2 . 12 2/5 Joint Distribution 3/5 Joint p.d.f of 2 continuous random variables Definition Joint p.d.f.. The joint probability density function of random variables X1 and X2 is a function f (x1 , x2 ) that possesses the properties 2 f (x1 , x2 ) ≥ 0; R +∞ R +∞ −∞ −∞ f (x1 , x2 )dx1 dx2 = 1; 3 P(a1 ≤ X1 ≤ b1 , a1 ≤ X2 ≤ b2 ) = 1 R b2 R b1 a2 a1 f (x1 , x2 )dx1 dx2 . Definition Marginal p.d.f.. The maginal probabiliy density functions of random variables X1 and X2 are respectively Z +∞ Z +∞ f1 (x1 ) = f (x1 , x2 )dx2 , and f2 (x2 ) = f (x1 , x2 )dx1 . −∞ −∞ 3/5 Joint cumulative distribution function and conditional p.d.f. Definition Joint cumulative distribution function. The joint cumulative distribution function is Z x2 Z x1 F (x1 , x2 ) = f (u, v )dudv = P(X1 ≤ x1 , X2 ≤ x2 ), −∞ −∞ so f (x1 , x2 ) = ∂ 2 F (x1 , x2 ) . ∂x1 ∂x2 Definition Conditional p.d.f.. The conditional probabiliy density of random variable X1 1 ,x2 ) being x1 given that random variable X2 being x2 is f (x1 |x2 ) = f f(x2 (x 2) provided that f2 (x2 ) 6= 0. 4/5 Independent random variable Definition Independence. Random variables X1 and X2 with p.d.f. f1 (x1 ) and f2 (x2 ) respectively are independent if and only if f (x1 , x2 ) = f1 (x1 )f2 (x2 ). If X1 and X2 are independent, we have f1 (x1 |x2 ) = f1 (x1 )f2 (x2 ) f (x1 , x2 ) = = f1 (x1 ). f2 (x2 ) f2 (x2 ) f2 (x2 |x1 ) = f (x1 , x2 ) f1 (x1 )f2 (x2 ) = = f2 (x2 ). f1 (x1 ) f1 (x1 ) Similarly, 5/5