16. 322 Stochastic Estimation and Control, Fall 2004 Prof. VanderⅤelde P(a<zsb)=aJJ、(xyx df(x)∫(y Let z=x+y, d=d 」女f(x)f(2-x)d f,(x),(z-x)dxd= This is true for all a b. Therefore f (=)=f(x)f,(z-x)dx =f((=-y)dy This result can readily be generalized to the sum of more independent random variables z=X1+X2+…+Xn f()=∫∫-Jdn2(x)(x)J(x)(-x-x Also, if w=Y-X, for X, Y independent f(w)=f(x),(w+x)dx ∫()/(-m Direct determination of the joint probability density of several functions o several ra andom variables Suppose we have the joint probability density function of several random variables x, Y, Z, and we wish the joint density of several other random variables defined as functions xyz U=u(X,Y, z V=v(X,Y,Z) W=(X,Y,2Z) Page 2 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde ∞ b x − ( < ≤ b) = dx f , P a Z (x y dy ∫ ∫ xy , ) −∞ a x − ∞ b x − dx f x () f y dy ∫ = ∫ x y ( ) −∞ ax − Let zx = + y dz , = dy ∞ b dx f x ∫ = () f z ( ) − x dz ∫ x y −∞ a b ∞ ⎡ ⎤ ⎢ f () x f z − x dx dz y ( ) = ∫ ∫ x ⎥ a ⎣−∞ ⎦ This is true for all a,b. Therefore: ∞ f () z = f x f z − x dx ∫ x ( ) y ( ) z −∞ ∞ f () ( ) y = y f z − y dy ∫ x −∞ This result can readily be generalized to the sum of more independent random variables. Z = X1 + X2 + ... + Xn ∞ ∞ ∞ f (z) = dx1 dx2... ∫ dxn−1 f (x f (x )... f (xn−1) f xn (z − x − x2 − ... − x ) x1 xn−1 z ∫ ∫ 1 1 n−1 ) x2 2 −∞ −∞ −∞ Also, if W Y = − X , for X,Y independent: ∞ f () w = f x f w + x dx ∫ x ( ) y ( ) w −∞ ∞ f () ( ) y = y f y − w dy ∫ x −∞ Direct determination of the joint probability density of several functions of several random variables Suppose we have the joint probability density function of several random variables X,Y,Z, and we wish the joint density of several other random variables defined as functions X,Y,Z. =  U u X Y Z ( ,, ) V v X Y Z = )  ( ,, = i W w X Y Z ( ,, ) Page 2 of 10