Expectation Shift Under utility theory S=Ey(x)-y(E(x)← S is the only difference between probabilistic and deterministic design VE(r) E(r(x)/\ vIX E(x) x
Expectation Shift x y (x ) E(x) y ( E (x)) E (y (x)) S fx f (x ) y (y(x)) S=E (y ( x))- y ( E ( x)) Under utility theory, S is the only difference between probabilistic and deterministic design
Probability distribution of sums If z is the sum of two random variables x and z=x+ Then the probability density function of z can be computed by convolution (2)=x=-)yA 00
Probability Distribution of Sums • If z is the sum of two random variables x and y • Then the probability density function of z can be computed by convolution z x y ³ f z z p ( z ) x ( z ] ) y (] ) d]
Convolution P:(2)=x(2-5)y()dk steps dff widths
Convolution ³ f z z p ( z ) x ( z ] ) y (] ) d]
Convolution g(t-3) P (2)=x(2-5)ys)ds step and ramp
Convolution ³ f z z p ( z ) x ( z ] ) y (] ) d]
Central Limit theorem The mean of a sequence of n iid random variables with Finite u Ex, -E(x;)o] 008>0 approximates a normal distribution in the limit of a large n
Central Limit Theorem The mean of a sequence of n iid random variables with – Finite P – approximates a normal distribution in the limit of a large n. ( ) < 0 2 f ! G G i i E x E x