16.322 Stochastic Estimation and Control, Fall 2004 Prof. VanderⅤelde d P(0 dr"=j/()("/ dx p(0)=∫x(x)dx=X ()=1++2()X2+…+()x+ The coefficients of the expansion are given by the moments of the distribution Thus the characteristic function can be determined from the moments Similarly the moments can be determined from the characteristic function ly by 1d"p() dt or by expanding o(r)into its power series in some other way and identifying the coefficients of the various powers of t. The Generating Function The generating function has its most useful application to random variables which take integer values only. Examples of such would be the number of telephone calls into a switchboard in a certain time interval, the number of cars entering a toll station in a certain time interval, the number of times a 7 is thrown in n tosses of 2 dice, etc For integer-valued random variables, the Generating Function yields the same advantages as the Characteristic Function and is of simpler form. Consider a random variable which takes the integer values k P(X=k=p (k=0,1,2…) For a discrete distribution you can sum in lieu of integration. The Characteristic Function for this random variable is 0)=E[e]∑e"p Pk If we define a new variable s=e/,we have G(s)=∑P2s Page 2 of 61 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde d φ t n jtx fx n ( ) = ∞ ∫ ( )( jx) e dx n dt −∞ ∞ n n n n 0 ∫ φ ) ( ) ( ) = j xn f (x dx = j X −∞ n n n ( ) X t + ... + 1 φ(t) = + jXt + ( ) X t + ... 1 j 2 2 2 j 2! n! The coefficients of the expansion are given by the moments of the distribution. Thus the characteristic function can be determined from the moments. Similarly, the moments can be determined from the characteristic function directly by n n 1 d φ(t) X = n n j dt t=0 or by expanding φ( )t into its power series in some other way and identifying the coefficients of the various powers of t. The Generating Function The generating function has its most useful application to random variables which take integer values only. Examples of such would be the number of telephone calls into a switchboard in a certain time interval, the number of cars entering a toll station in a certain time interval, the number of times a 7 is thrown in n tosses of 2 dice, etc. For integer-valued random variables, the Generating Function yields the same advantages as the Characteristic Function and is of simpler form. Consider a random variable which takes the integer values k: PX ( = k ) = p (k=0,1,2,…) k For a discrete distribution you can sum in lieu of integration. The Characteristic Function for this random variable is ∞ jtk ( ) = E ⎡e p jtX φ t ⎤ = ∑e ⎣ ⎦ k k =0 ∞ = ∑ p e jt k k =0 k ( ) If we define a new variable s e jt = , we have ∞ Gs k k ( ) = ∑ p s k =0 Page 2 of 6