16. 322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Lecture 6 Example: Sun of two independent randoin variable Z=X+Y f(=xh=P(a<Z≤b) =P(a<X+Y≤b P(a-X<Y≤b-X) =im∑P(x<X≤x+d)P(a-x<y≤b-x) =im∑f(x)dxf(y f(x)x」f(y)小y We can reach this same point by just integrating the joint probability density function for X and y over the region for which the event is true In the interior strip, the event a <=s bis true. Page 1 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Lecture 6 Example: Sum of two independent random variables Z=X+Y b f z dz () ( = Pa < Z ≤ b) ∫ z a = P a( < X +Y ≤ b) = P a( − X < Y ≤ b − X ) = lim ∑P x( < X ≤ x + dx P a ) ( − x < Y ≤ b − x) dx→0 x b x − = lim ∑ f ( x dx f y dy ) ( ) dx→0 x ∫ y x a x − ∞ b x − () y f x dx ( ) ∫ = f y dy ∫ x −∞ a x − We can reach this same point by just integrating the joint probability density function for X and Y over the region for which the event is true. In the interior strip, the event a z < ≤ b is true. Page 1 of 10