independent and conditional

Independent random variables
Def :- the joint distribution r.vs X and Y are said to be independent if and only if their joint dis, fun , F(x,y) can be written as the product of the marginal dis, fun , F(X) and F(Y)
F(X,Y)= F(X).F(Y)
If X and Y are continuous r.v’s, with joint p.d.f. f(x,y). then
X and Y are independent if and only if f (x,y)=f(x).f(y).
If X and Y are discrete r.v’s with joint p.m.f p(x,y), then X and Y are independent iff
P(x,y)=p(x).p(y), for all x and y.

Example:- let X and Y be two r.v’s having the joint p.d.f.

f(x,y)=4e^(-2(y+x)) x>0 ,y>0

The marginal density fun, of X and of Y are respectively
f(x)=?_0^???4e^(-2(y+x)) ? dy =2e^(-2x) , x>0
And
f(y)=?_0^???4e^(-2(y+x)) ? dx =2e^(-2y) , y>0
Then
f(x) .f(y)=2e^(-2x) 2e^(-2y)=4e^(-2(x+y))=f(x,y)
Therefore X and Y are independent r.v’s.
Example:- :- let X and Y are two r,vs with j.p.m.f given by
Y


X
0 1 Total p(x)
0 4/9 2/9 6/9
1 2/9 1/9 3/9
Total p(y) 6/9 3/9 1
We have
P_x (0).P_y (0)=6/9.6/9=4/9=p(0,0)
P_x (1).P_y (0)=3/9.6/9=2/9=p(1,0)
P_x (0).P_y (1)=6/9.3/9=2/9=p(0,1)Independent random variables
Def :- the joint distribution r.vs X and Y are said to be independent if and only if their joint dis, fun , F(x,y) can be written as the product of the marginal dis, fun , F(X) and F(Y)
F(X,Y)= F(X).F(Y)
If X and Y are continuous r.v’s, with joint p.d.f. f(x,y). then
X and Y are independent if and only if f (x,y)=f(x).f(y).
If X and Y are discrete r.v’s with joint p.m.f p(x,y), then X and Y are independent iff
P(x,y)=p(x).p(y), for all x and y.

Example:- let X and Y be two r.v’s having the joint p.d.f.

f(x,y)=4e^(-2(y+x)) x>0 ,y>0

The marginal density fun, of X and of Y are respectively
f(x)=?_0^???4e^(-2(y+x)) ? dy =2e^(-2x) , x>0
And
f(y)=?_0^???4e^(-2(y+x)) ? dx =2e^(-2y) , y>0
Then
f(x) .f(y)=2e^(-2x) 2e^(-2y)=4e^(-2(x+y))=f(x,y)
Therefore X and Y are independent r.v’s.
Example:- :- let X and Y are two r,vs with j.p.m.f given by
Y


X
0 1 Total p(x)
0 4/9 2/9 6/9
1 2/9 1/9 3/9
Total p(y) 6/9 3/9 1
We have
P_x (0).P_y (0)=6/9.6/9=4/9=p(0,0)
P_x (1).P_y (0)=3/9.6/9=2/9=p(1,0)
P_x (0).P_y (1)=6/9.3/9=2/9=p(0,1)
P_x (1).P_y (1)=3/9.3/9=1/9=p(1,1)

P_x (1).P_y (1)=3/9.3/9=1/9=p(1,1)