16.322 Stochastic Estimation and Control, Fall 2004 Prof. VanderⅤelde For the density function F F ax, ax,.xk nfx-x(4…,n) Rk,uk Marginal density If you integrate above over all variables but one, it is referred to as the marginal ∫d,f,-(x,,x,) n-I terms: all except r, Mutually independent sets of random variable Definition of independence P[X∈s,X2∈S2,]=P[X∈s]P[X2∈s2] for any sets The product rule holds for joint probability distribution and density functions for independent random variables F12x(x1,x2,x3…)=F2(x)F2(x2)F(x2) )=f2(x)2(x2)2(x2) Expectations E[8(x)]=dx ax,g(x)/(x) Page 5 of 616.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde For the density function: ∂k f x1 ,..., xk ( x1,..., x ) = x x2...∂xk Fx1 ,..., x ( x1,..., xk ) k k ∂ ∂1 ∂k = x x2...∂xk Fx1,..., x ( x1,..., xk , ∞,..., ∞) n ∂ ∂1 x1 xk ∞ ∞ ∂k du1... ∫ duk ∫ duk+1 = ... du f (u ,..., u ) ∂ ∂ ∫ ∫ n x 1,..., x 1 n n x x2...∂x 1 k −∞ −∞ −∞ −∞ ∞ ∞ ∫ n x 1,...,x ( x1,..., x uk+1 du ,..., u ) k+1... du f k , = ∫ n n −∞ −∞ ∞ ∞ = ∫ duk+1... ∫ du f ( x1,..., x ) n x 1,..., x n n −∞ −∞ Marginal density If you integrate above over all variables but one, it is referred to as the marginal density. ∞ ∞ f xi ( xi) = dx1... dx f ,..., x ( x ,..., x ) ∫ ∫ n x1 n 1 n −∞ 144244−∞ 3 n-1 terms: all except xi Mutually independent sets of random variables Definition of independence: P X1 ∈ s X ∈ s2 ,... ] = PX ∈ s P X [ 2 ∈ s ]... 1, 2 [ 1 1 ] [ 2 for any sets s1, s2, … The product rule holds for joint probability distribution and density functions for independent random variables. F , ( x x x ,...) = F x F x F x 2 )... x x x , 3 ,... 1, 2 , 3 x1 ( 1) x2 ( 2 ) x ( 12 2 f ( x x x ,...) = f ( x f x2 ( x f ( x )... x x x , , 3 ,... 1, 2 , 3 x1 1) 2 ) x2 2 1 2 Expectations ∞ ∞ Egx [ ( )] = dx ( ) ( ) 1... dx g x f x ∫ ∫ n −∞ −∞ Page 5 of 6