Histograms of random variables 100.0 00527178070218AN2 andom variable x 03/1203 12540Lec10 Characterization random variables When the probability distribution is known, the following statistical descriptions are used for random variable x with density function f(x) Expected value <h(x)> h(x)f(x)a Expectation <r> xf (x)dx=u Variance <(x-H)2>(x-u2'f(x)dx Square root of variance is called standard deviationHistograms of random variables 200.0 Gaussian Uniform 490/sqrt(2pi)*exp(-x^2/2) 150.0 100.0 Number of samples 50.0 0.0 -3.75 -2.75 -1.75 -0.75 0.25 1.25 2.25 3.25 Random Variable x 03/12/03 12.540 Lec 10 15 03/12/03 12.540 Lec 10 16 Characterization Random Variables Expected Value < h(x) > Ú h(x) f (x)dx Expectation < x > Ú xf (x)dx = m Variance < (x - m) 2 > (x - m) 2 Ú f (x)dx • When the probability distribution is known, the following statistical descriptions are used for random variable x with density function f(x): Square root of variance is called standard deviation 8