Perluasan ke n buah variabel random

1. Perluasanke n buahvariabel random

5. Definisi • Misalkansuatupercobaan random denganruangsampelC • Misalkan Xi memetakanuntuksetiap c anggotaC kesatudanhanyasatuXi(c)=xi : i=1,2,..,n • Ruangdarivariabel-variabel random tersebutadalahA= C • Selanjutnyamisalkan A  A, maka : dimana C={c : c Cdan

6. f memenuhipdfapabila: 1. f 2. FungsiDistribusidari n variabel random :

7. Marginal distributions

8. Definition Let X1, X2, …, Xq, Xq+1…, Xk denote k discrete random variables with joint probability function p(x1, x2, …, xq, xq+1 …, xk ) then the marginal joint probability function of X1, X2, …, Xq is

9. Definition Let X1, X2, …, Xq, Xq+1…, Xk denote k continuous random variables with joint probability density function f(x1, x2, …, xq, xq+1 …, xk ) then the marginal joint probability function of X1, X2, …, Xq is

11. Conditional distributions

12. Definition Let X1, X2, …, Xq, Xq+1…, Xk denote k discrete random variables with joint probability function p(x1, x2, …, xq, xq+1 …, xk ) then the conditional joint probability function of X1, X2, …, Xq given Xq+1 = xq+1 , …, Xk= xkis

13. Definition Let X1, X2, …, Xq, Xq+1…, Xk denote k continuous random variables with joint probability density function f(x1, x2, …, xq, xq+1 …, xk ) then the conditional joint probability function of X1, X2, …, Xq given Xq+1 = xq+1 , …, Xk= xkis

14. Definition Let X1, X2, …, Xq, Xq+1…, Xk denote k continuous random variables with joint probability density function f(x1, x2, …, xq, xq+1 …, xk ) then the variables X1, X2, …, Xq are independent of Xq+1, …, Xkif A similar definition fordiscrete random variables.

15. Definition Let X1, X2, …, Xk denote k continuous random variables with joint probability density function f(x1, x2, …, xk ) then the variables X1, X2, …, Xk are called mutuallyindependent if A similar definition fordiscrete random variables.

16. Example Let X, Y, Z denote 3jointly distributed random variable with joint density function then Find the value of K. Determine the marginal distributions of X, Y and Z. Determine the joint marginal distributions of X, Y X, Z Y, Z

17. Solution Determining the value of K.

18. The marginal distribution of X.

19. The marginal distribution of X,Y.

20. Find the conditional distribution of: • Z given X = x, Y = y, • Y given X = x, Z = z, • X given Y = y, Z = z, • Y , Z given X = x, • X , Z given Y = y • X , Y given Z = z • Y given X = x, • X given Y = y • X given Z = z • Z given X = x, • Z given Y = y • Y given Z = z

21. The marginal distribution of X,Y. Thus the conditional distribution of Z given X = x,Y = y is

22. The marginal distribution of X. Thus the conditional distribution of Y , Z given X = x is

23. Definition Let X1, X2, …, Xn denote n jointly distributed random variable with joint density function f(x1, x2, …, xn ) then

24. Example Let X, Y, Z denote 3jointly distributed random variable with joint density function then Determine E[XYZ].

25. Solution:

26. Some Rules for Expectation

27. Thus you can calculate E[Xi] either from the joint distribution of X1, … , Xn or the marginal distribution of Xi. Proof:

28. The Linearity property Proof:

29. (The Multiplicative property)Suppose X1, … , Xq are independent of Xq+1, … , Xk then In the simple case when k = 2 if X and Y are independent

30. Proof:

32. Some Rules for Variance

33. Proof Thus

34. Note: If X and Y are independent, then

35. Definition Let X1, X2, … Xkbe a jointly distributed random variables (discrete or continuous), then the joint moment generating function is defined to be:

36. Definition Let X1, X2, … Xkbe a jointly distributed random variables (discrete or continuous), then the joint moment generating function is defined to be: