Help on to find probability density function

[musademirtas]

hey guys, i am really confused on something.here is the thing:
i have;

i=x+(x^2-y)^(1/2)

and here x is uniform distribution on (a,b)
y is uniform distribution on (c,d)
x and y independent
i need to find the probability density function of i but how?
actually i don't know how to start!

 

[micromass]

Hi musademirtas!

First you will need to know the joint distribution of X and Y. This is easy because of independence:

$$f_{X,Y}(x,y)=f_X(x)f_Y(y)$$

Now, to find the pdf, you will need to find the cdf first. That is, for each a, you will want to calculate

$$P\{X+\sqrt{X^2-Y}\leq a\}=\iint_{\{(x,y)~\vert~x+\sqrt{x^2-y}\leq a\}}{f_{X,Y}(x,y)dxdy}$$

To evaluate this integral, you'll need to know the region $\{(x,y)~\vert~x+\sqrt{x^2-y}\leq a\}$ somewhat better.

So I suggest you first find out for which tuples the equality holds. That is, for which x and y does it hold that

$$x+\sqrt{x^2-y}=a$$

(hint: the answer will be a straight line!)

 

[musademirtas]

hi micromass
thanks for the help but can you solve it? because i used your help to solve it but i couldn't do it.

 

[micromass]

Well, first you're going to need to write

$$x+\sqrt{x^2-y}=a$$

in function of y. What do you get for that?

 

[musademirtas]

ok.i got y=2ax-x^2 then what? how am i going to use this?