1. The joint probability distribution function of X and Y is denoted by f XY (x,y). The marginal probability distribution function of X, f X (x) is obtained by i. summing the probabilities corresponding to each y value and the given x. (discrete case) f X (x)=  Y f XY (x,y)|X=x Joint Probability Distribution

2. The conditional probability distribution function of X given Y is denoted by f X|Y (x|y). f X|Y (x|y)=f X, Y (x,y)/ f Y (y) [We similarly define f Y|X (y|x)] Random variables X and Y are independent if and only if f X|Y (x|y)=f X (x) for all x and y ii. Integrating out Y from the joint pdf (continuous case) f X (x)=  Y f XY (x,y)dy

3. Joint Probability Distribution Covariance of two random variables X and Y Cov(X,Y)  E{(X-  X )(Y-  Y )}  E(XY- Y  X –X  Y +  X  Y )  E(XY) -  X E(Y) –  Y E(X) +  X  Y  E(XY)–  X  Y -  X  Y +  X  Y  E(XY)-  X  Y

4. The Coefficient of Correlation between two random variables X and Y  (X,Y)  Cov(X,Y)/  X  Y Two random variables X and Y are uncorrelated if  (X,Y)  0; or if E(XY)=  X  Y

5. An important result: Suppose that X and Y are two random variables that have mean  X,  Y and standard deviation  X and  Y respectively. Then for Z  aX +bY E(Z)  a  X + b  Y VarZ  a 2 VarX + b 2 VarY + 2abcov(X,Y)  a 2   X + b 2   Y + 2ab  XY  X  Y

6. VarZ  a 2 VarX + b 2 VarY + 2abcov(X,Y)  a 2   X + b 2   Y + 2ab  XY  X  Y where  XY  Coefficient of Correlation between X and Y. If  XY = -1 then VarZ = (a  X -b  Y ) 2

7. Joint Probability Distribution Recall: Two random variables x and y are independent if their joint p.d.f. f XY (x,y) is the product of the respective marginal p.d.f. f X (x) and f Y (y). That is, f XY (x,y) = f X (x). f y (y)

8. Theorem: Independence of two random variables X and Y imply that they are uncorrelated (but the converse is not always true)

9. Proof: E(XY) =  xy f XY (x,y)dxdy E(XY) =  xy g(x)h(y)dxdy E(XY) =  (  x g(x)dx)y h(y)dy E(XY) =  (  X )y h(y)dy E(XY) =  X  y h(y)dy E(XY) =  X  Y

10. The Normal Distribution A continuous distribution with the pdf: f(X) = {1/(  } e –1/2[(X-  x )/  x )2 ] For the Standard Normal Distribution Variable Z,  f(z) = {1/  } e –(1/2) z 2

11. Suppose that X and Y are two random variables such that they have mean  X,  Y and standard deviation  X and  Y respectively. Also assume that both X and Y are normally distributed. Then if W  aX +bY

12. W ~ Normal(  w,   w ) with  w  a  X + b  Y and   w  a 2   X + b 2  X + 2ab  XY  X  Y where  XY is the relevant correlation coefficient.

13. Message: A linear combination of two or more normally distributed (independent or not) r.v.s has a normal distribution as well.

14. The   distribution: Consider Z ~ Normal(0,1). Consider Y = Z 2. Then Y has a   dd istribution of 1 degree of freedom (d.o.f.). We write it as Y ~   .

15. Consider Z 1, Z 2, …Z n independent random variables each ~ Normal(0,1) Then their sum  Z i 2 has a   distribution with d.o.f. = n. That is,  Z i 2 ~   (n)

16. Consider two independent random variables X ~ Normal(0,1) and Y ~   (n) Then the variable w  X/  (Y/n) has a t-distribution with d.o.f. = n. That is, w ~ tt (n)

17. An Application Then the variable w  (X MEAN –  s/  n) hh as a t-distribution with d.o.f. = n-1 where s is an unbiased estimator of  Consider X ~ (   ). Then X MEAN ~ Normal (   /n) if n is ‘large’ (CLT) Consider X ~ Normal(   ). Then X MEAN ~ Normal (   /n)

18. Suppose that X ~   (m). and Y ~  (n) and the variables X and Y are independent. V ~ F m,n Then the variable v ≡ (x/m) /(y/n) has an F distribution with the numerator d.o.f. = m and the denominator d.o.f = n.

19. Suppose that X ~  (1). and Y ~  (n) and the variables X and Y are independent. Then the variable v ≡ x /(y/n) has an F distribution with the numerator d.o.f. = 1 and the denominator d.o.f = n. V ~ F 1,n Clearly,  V ~ t (n)