16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde which is called the Generating Function. It has all the interesting properties of the characteristic function. Note that [>0 corresponds to s>1 Let s establish the connection between moments of a distribution and the =2 d-g k(k-Dp Prs kp Just calculate ds andd and reorganize them in terms of X and X =∑=X,←1" moment expression d2G dG ∈2- moment expression Each moment is a linear combination of its order derivative and lower order derivatives. The generating function for the sum of independent integer-valued variables is the product of their generating functions. This is harder to prove han the same property of the characteristic function, but it does, in fact, hold Multiple randon variables Characterizing a joint set of random variables, define a probability distribution function F(x)=P(X1≤x12X2≤x2,Xn≤xn) This is called the joint probability distribution function Properties: If any of the arguments xi goes to -oo, then F(x)>0 lim F(x=0 Page 3 of 616.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde which is called the Generating Function. It has all the interesting properties of the characteristic function. Note that t → 0 corresponds to s →1. Let’s establish the connection between moments of a distribution and the generating function: ∞ dG = ∑kp sk −1 ds k =0 k 2 ∞ ( dG k −2 = ∑kk −1) p s k ds2 k =0 ∞ ∞ 2 k −2 k −2 = ∑k p s −∑kp s k k k=0 k=0 2 dG d G Just calculate and and reorganize them in terms of X and X 2 : ds ds2 s=1 s=1 ∞ dG st = ∑kp = X , ← 1 moment expression k ds s=1 k =0 2 ∞ ∞ dG 2 = ∑kp −∑kp k k ds2 k =0 s=1 k =0 2 X 2 dG dG = + ← 2nd moment expression ds2 ds s=1 s=1 Each moment is a linear combination of its order derivative and lower order derivatives. The generating function for the sum of independent integer-valued variables is the product of their generating functions. This is harder to prove than the same property of the characteristic function, but it does, in fact, hold true. Multiple Random Variables Characterizing a joint set of random variables, define a probability distribution function F( ( x) = P X ≤ x , X ≤ x2 ,..., X ≤ x ) 1 1 2 n n This is called the joint probability distribution function. Properties: If any of the arguments xi goes to −∞ , then F() x → 0 . lim F(x) = 0 any xi→−∞ Page 3 of 6