4. Solve 4.34 e) Find the expected value of XY. f) Find the probability distribu

Question

4. Solve 4.34 e) Find the expected value of XY. f) Find the probability distribution ofX given Y-2 Answer 5. Solve 4.50 g) Find the expected value of X given Y is 2: Answer 6. Given two random variables, X and Y, table below illustrates their joint distribution. a) Find the marginal distributions of X and Y. h) Find covariance between X and Y 01 .2 1 12 2 1 210 0 ht 1) Find correlation between X and Y b) Find the expected values of X andY EY c) Find variance of X j) Are X and Y highly correlated? Comment. Va(X) = d) Find variance ofY k) Fill in the blanks: E(14X)- E(-120X-100)- E(300) 300 Var(141) = Var(-120X-100)- Var(300) = Var(Y) =

Explanation / Answer

from above joint distribution table:

below is the marginal pdf of X

and marginal pdf of Y:

from above:

b)E(X)=0.7 ; E(Y) =0.8

c) Var(X) =0.41

d) Var(Y)=0.56

e) E(XY) =0.4

f)pdf of X given Y =2:

g)E(X|Y=2) =0*0.5+1*0.5+2*0=0.5

h)Cov(X,Y)=E(X Y)-E(X)*E(Y) =-0.16

i)Corr(X,Y) =Cov(X,Y)/((Var(X)*Var(Y))1/2 =-0.3339

j)No as correlation coefficient is lower then 0.5.

k)E(14X)=14*E(X)= 9.8

Var(14X) =142Var(X) =80.36

E(-120X-100)=-120E(X)-100=184

Var(-120X-100)=1202Var(X) =5904

E(300)=300 ; Var(300) =0

y x 0 1 2 Total 0 0.1000 0.2000 0.1000 0.4000 1 0.2000 0.2000 0.1000 0.5000 2 0.1000 0.0000 0.0000 0.1000 Total 0.4000 0.4000 0.2000 1.0000

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