16322 Stochastic Estimation and Control. Fall 2004 Prof vander velde If all of the xi go to oo, then F(x)>1 lim FO F(x)is monotonically non-decreasing in each x, Define joint density function by differentiation f(x) f(x)≥0,x F(x…x)=」dn…Jmnf1=,(a1-n) Setting each x→∞, du F,(x1…,x)=P(1≤x,…,Xn≤x) =P(X1≤x,…,Xk≤xk,Xk1≤∞,…,Xn≤∞) Page 4 of 616.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde If all of the xi go to ∞, then F( ) x →1. lim F(x) = 1 all xi→∞ Fx ( )is monotonically non-decreasing in each xi. Define joint density function by differentiation: n f ( ) x = ∂ x x2 ∂ ∂ ...∂x 1 n f ( ) x ≥ ∀0, x xn x1 Fx1... xn ( x1...x ) = ∫ du1... ∫ du f (u ...u ) n n x1... x 1 n n −∞ −∞ Setting each x → ∞ , i ∞ ∞ du1... du F (u ,..., u ) = 1 ∫ ∫ n u1 ,..., u 1 n n −∞ −∞ F ( x1,..., xk ( x1,..., xk ) = P X 1 ≤ x1,..., Xn ≤ xn ) = P X( 1 ≤ x ,..., Xk ≤ x Xk+1 ≤ ∞,..., X ≤ ∞) 1 k , n = Fx1,..., x ( x1,..., x , ∞,..., ∞) k n Page 4 of 6