Simple stochastic models 2

1. Simple stochastic models 2

2. Continuous random variable X X – can take values: - < x < + Cumulative probability distribution function: PX(x) = P(X  x) Probability density function:

3. Normal distribution X ~ normal(,), E(X)= , V(X)=  2 pdf: cpdf:

4. =0, =1 0.4 0.3 pdf 0.2 0.1 0 -4 -3 -2 -1 0 1 2 3 4 cpdf x

5. Central limit theorem Y = X1 + X2 + … +Xn Xi - independent, zero mean, equal varianceV If n is large then: ~ normal(0,1)

6. 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 Example – students body heights Histogram of students’ body heights versus normal probability density function 160 165 170 175 180 185 190 195 200 205

7. 9 8 7 6 5 4 3 2 1 0 Example – children birth weights Histogram of children birth weights versus normal probability density function -4 x 10 0 1000 2000 3000 4000 5000 6000

8. Binomial becomes normal 0.12 0.1 0.08 0.06 0.04 0.02 0 0 5 10 15 20 25 30 35 40 45 50 o – binomial(0.5,50) normal(25,3.5355)

9. How do we fit normal distribution to data ? Data: X1, X2, …, Xn

10. How do we estimate parameters of distributions using data ? • How do we verify that data follow a given distribution ?

11. Characteristic function X – with pdf p(x) characteristic function:

12. Properties

13. Properties Y=aX+b, a,b - constants

14. Characteristic function of normal distribution X ~ normal(,),

15. Continuity theorem for characteristic functions

16. Two dimensional distributions X, Y Probability density function: p(x,y) Cumulative pdf:

17. Independent random variables X, Y independent pXY(x,y)=pX(x) pY(y) Convolution integral Z=X+Y pZ = pX * pY

18. Convolution and characteristic functions Z=X+Y

19. Use of characteristic functions to prove Central Limit Theorem Y = X1 + X2 + … +Xn i=1,2…,n so: and