1. ## Bivariate Rv help please

The joint pdf of a bivariate Rv (X,Y) is given by

Fxy(x,y)={k(x+2y),0<x<y<1
0otherwise

K a constant

Find
1. Obtain value for k
2. Find marginal pdfs of X and Y
3. Obtain the conditional density of X given Y=y, What is this density when Y=0.2
4. Are X and Y independent

2. Originally Posted by kermit
The joint pdf of a bivariate Rv (X,Y) is given by

Fxy(x,y)={k(x+2y),0<x<y<1
0otherwise

K a constant

Find
1. Obtain value for k
2. Find marginal pdfs of X and Y
3. Obtain the conditional density of X given Y=y, What is this density when Y=0.2
4. Are X and Y independent
The support is the region of the xy-plane enclosed by the lines x = 0, y = 1 and y = x. Did you draw a sketch? So, for example:

1. Solve $\int_{x = 0}^{x = 1} \int_{y=x}^{y = 1} f(x, y) \, dy \, dx = 1$ for k.

The other questions are done using this support and applying the appropriate definitions/formula. If you need more help, please show all your work and say where you are stuck.

3. thanks for getting back so quick.

See the attachment, this is how far i got im not sure if its correct can you have a look and let me know if i went wrong.

thanks

kermit

4. Originally Posted by kermit
thanks for getting back so quick.

See the attachment, this is how far i got im not sure if its correct can you have a look and let me know if i went wrong.

thanks

kermit
Many people don't open attachments. It's better if you try to learn some basic latex (see my signature) and type your work in the post.

5. Originally Posted by kermit
The joint pdf of a bivariate Rv (X,Y) is given by

Fxy(x,y)={k(x+2y),0<x<y<1
0otherwise

K a constant

Find
1. Obtain value for k
2. Find marginal pdfs of X and Y
3. Obtain the conditional density of X given Y=y, What is this density when Y=0.2
4. Are X and Y independent
1. i think ur right
2. For mdf of x, integrate from x to 1 for dy, for mdf of y, integrate from 0 to y for dx
3. Conditional density of X given Y=y is f(x,y)/f(y) and do same for Y=0.2
4. X and Y are independent IFF f(xy)=f(x)f(y)--- mdf's

6. Ok heres how far i have gotten, im just wondering if it is integrated correctly or have i gone wrong anywhere.

$
\int_{x=0}^{x=1}\int_{y=x}^{y=1}k(x+2y) \ dy \ dx=1
$

$
k\int_{x=0}^{x=1}(xy+y^2)_{x}^{1} \ dx=1
$

$
k\int_{x=0}^{x=1}(x+1)-(x^2+x^2) \ dx=1
$

$
k((\dfrac{x^2}{2}+x)-(\dfrac{2x^3}{3}))_{0}^{1} \ dx=1
$

$
k((\dfrac{1}{2}+1)-(\dfrac{2}{3}))=1
$

7. Originally Posted by firebio
2. For mdf of x, integrate from x to 1 for dy, for mdf of y, integrate from 0 to y for dx
for the mdf of x i have intergrated from x to 1 for dy. as you mentioned above

but for the mdf of y i intergrated from 0 to 1?? for dx am i wrong and if so why is it 0 to y could you please explain

8. Originally Posted by kermit
Ok heres how far i have gotten, im just wondering if it is integrated correctly or have i gone wrong anywhere.

$
\int_{x=0}^{x=1}\int_{y=x}^{y=1}k(x+2y) \ dy \ dx=1
$

$
k\int_{x=0}^{x=1}(xy+y^2)_{x}^{1} \ dx=1
$

$
k\int_{x=0}^{x=1}(x+1)-(x^2+x^2) \ dx=1
$

$
k((\dfrac{x^2}{2}+x)-(\dfrac{2x^3}{3}))_{0}^{1} \ dx=1
$

$
k((\dfrac{1}{2}+1)-(\dfrac{2}{3}))=1
$
It looks OK.

Originally Posted by kermit
for the mdf of x i have intergrated from x to 1 for dy. as you mentioned above

but for the mdf of y i intergrated from 0 to 1?? for dx am i wrong Mr F says: Yes. Read below.

and if so why is it 0 to y could you please explain
The support is the region of the xy-plane enclosed by the lines x = 0, y = 1 and y = x. Did you draw a sketch?
Furthermore, you were explicitly told what to do in post #5. Have you been taught how to integrate over a region?

9. thanks for your help, i have been given a crash coursein it, and cant seem to figure out how to sketch out the region, therfore i have a problem with getting the limits, any help you can give in how to sketvh out this problem would be helpful

10. Originally Posted by kermit
thanks for your help, i have been given a crash coursein it, and cant seem to figure out how to sketch out the region, therfore i have a problem with getting the limits, any help you can give in how to sketvh out this problem would be helpful
Draw the line y = x. Draw the line y = 1. Draw the line x = 1. x < y is satisfied when you're below the line y = x. This gives the region I described in my first reply.

11. You may want to try to reverse the order of integration.
The algebra is a bit easier...

$1=k\int_0^1\int_0^y(x+2y)dxdy$

$f_X(x)=\int_x^1f(x,y)dy$

$f_Y(y)=\int_0^yf(x,y)dx$

The conditional density of X given Y is ${f(x,y)\over f_Y(y)}$ then set y=.2 for the next question.

X and Y are dependent since the support has x<y, but if you calculate the two marginals

then you will see that $f(x,y)\ne f_X(x)f_Y(y)$