16.322 Stochastic Estimation and Control, Fall 2004 Prof. VanderⅤelde To get the density for the product only, integrate out with respect to v. f(u)=言f,(v,=)a If X and Y are independent this becomes (0-J f() f(x)f The Uniform distribution In our problems we have been using the uniform distribution without having concisely defined it. This is a continuous distribution in which the probability density function is uniform(constant) over some finite interval F(x) 1/(b-a) Thus a random variable having a uniform distribution takes values only over some finite interval(a, b) and has uniform probability density over that interval In what situation does it arise? Examples include part tolerances, quantization error, limit cycles. Often you do not know anything more than that the unknown value lies between known bounds Page 4 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde To get the density for the product only, integrate out with respect to v. ∞ () = ∫ 1 f u f x y (v, u )dv u , u v −∞ If X and Y are independent this becomes ∞ 1 u fu () = ∫ ( ) ⎛ ⎞ f v f y ⎜ ⎟dv, or u x v ⎝ ⎠ v −∞ ∞ 1 u f () u = ∫ ( ) ⎛ ⎞ f x f y ⎜ ⎟dx u x x ⎝ ⎠ x −∞ The Uniform Distribution In our problems we have been using the uniform distribution without having concisely defined it. This is a continuous distribution in which the probability density function is uniform (constant) over some finite interval. Thus a random variable having a uniform distribution takes values only over some finite interval (a,b) and has uniform probability density over that interval. In what situation does it arise? Examples include part tolerances, quantization error, limit cycles. Often you do not know anything more than that the unknown value lies between known bounds. Page 4 of 10