Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Simple stochastic models 2 Continuous random variable X X – can take values: - < x < + Cumulative probability distribution function: PX(x) = P(X x) Probability density function: P ( x X x x ) p( x) lim x 0 x Normal distribution X ~ normal(,), E(X)= , V(X)= 2 pdf: 1 ( x ) 2 /( 2 2 ) p ( x) e 2 1 cpdf: P( x) 2 x e ( u ) 2 /( 2 2 ) du =0, =1 0.4 0.3 pdf 0.2 0.1 0 -4 -3 -2 -1 0 1 2 3 4 -3 -2 -1 0 1 2 3 4 1 0.8 cpdf 0.6 0.4 0.2 0 -4 x Central limit theorem Y = X1 + X2 + … +Xn Xi - independent, zero mean, equal variance V If n is large then: Y nV ~ normal(0,1) Example – students body heights Histogram of students’ body heights versus normal probability density function 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 160 165 170 175 180 185 190 195 200 205 Example – children birth weights 9 Histogram of children birth weights versus normal probability density function x 10-4 8 7 6 5 4 3 2 1 0 0 1000 2000 3000 4000 5000 6000 Binomial becomes normal 0.12 0.1 0.08 0.06 0.04 0.02 0 0 5 10 15 o – binomial(0.5,50) 20 25 30 35 40 45 50 normal(25,3.5355) How do we fit normal distribution to data ? Data: X1, X2, …, Xn 1 n mean X i n i 1 1 n 2 std ( X mean ) i n 1 i 1 • How do we estimate parameters of distributions using data ? • How do we verify that data follow a given distribution ? Characteristic function X – with pdf p(x) characteristic function: ( ) e j x p( x)dx ( ) E e j X j 1 Properties ( 0) 1 ' (0) jE( X ) ' ' (0) E ( X 2 ) Properties X ( ) E e j X Y=aX+b, a,b - constants Y ( ) Ee j ( aX b ) e j b X (a ) Characteristic function of normal distribution X ~ normal(,), 1 ( x ) 2 /( 2 2 ) p ( x) e 2 ( ) e 2 / 2 2 j / Continuity theorem for characteristic functions Two dimensional distributions X, Y Probability density function: p(x,y) b1 b2 P(a1 X b1 , a2 Y b2 ) p( x, y )dxdy a1 a2 Cumulative pdf: x y P ( X x, Y y ) p(u, v)dudv Independent random variables X, Y independent pXY(x,y)=pX(x) pY(y) Convolution integral Z=X+Y pZ ( z ) p X ( z y ) pY ( y )dy p X ( x) pY ( z x)dx p Z = pX * p Y Convolution and characteristic functions X ~ X ( ) Y ~ Y ( ) Z=X+Y Z ( ) X ( )Y ( ) Use of characteristic functions to prove Central Limit Theorem Y = X1 + X2 + … +Xn Y nV ( ) E e j X so: Y ( ) ( ) n i i=1,2…,n ) and Y ( ) ( nV nV n V 1 2 ( ) 1 o 2nV nV n 1 2 n log ( ) 2 nV