pdf of function of two random variables (It is little difficult)

**kuywam** · November 23rd 2009, 04:16 AM

Hello
I am Ph.D. student studying electrical engineering.
I have a question related with cdf of function of two random variables.
The question is attached in this post as pdf file.

Please, give me an answer.

**Moo** · November 23rd 2009, 10:07 AM

Hello,

(a) is ... weird Oo

First thing : if X and Y are independent, you can write the product $f_X(x')f_Y(y')$
if not, you have to find the joint pdf of X and Y.

Second thing :
The whole formula is not very correct...

Assuming they're independent :

$\begin{aligned} P\left(\tfrac{X}{1+Y}\leq z\right)=P(X \leq z(1+Y)) &=P\left(Y\geq \tfrac{X-z}{z}\right) \\ &=\int_{x'=0}^\infty \int_{y'=0}^{\tfrac{x'-z}{z}} f_Y(y')f_X(x') ~dy' ~dx' \quad (*) \\ &=\int_{x'=0}^\infty f_X(x')\left(\int_{y'=0}^{\tfrac{x'-z}{z}} f_Y(y') ~dy'\right) ~dx' \\ &=\int_{x'=0}^\infty f_Y(y') F_Y\left(\tfrac{x'-z}{z}\right) ~dx \\ &=\mathbb{E}\left[F_Y\left(\tfrac{X-z}{z}\right)\right] \end{aligned}$

For the boundaries in (*), that's using the fact that $P\left(Y\geq \frac{X-z}{z}\right)=\mathbb{E}\left[\bold{1}_{Y\geq \frac{X-z}{z}}\right]$

and that for any measurable function h, $\mathbb{E}[h(X,Y)]=\iint h(x,y)f_X(x)f_Y(y) ~dx~dy$ if X and Y are independent.*

erm... good luck

How did you get your equality (a) ?

**Laurent** · November 23rd 2009, 11:35 AM

Originally Posted by kuywam

Hello
I am Ph.D. student studying electrical engineering.
I have a question related with cdf of function of two random variables.
The question is attached in this post as pdf file.

Please, give me an answer.

As an addition to Moo's post, I think what you were looking for is the following formula: (looks very much like (b), but this one is correct)

$P\left(\frac{X}{1+Y}\leq z\right)=$ $P(X\leq z(1+Y))=\int_0^{\infty} P(X\leq z(1+y))f_Y(y)dy = \int_0^\infty F_X(z(1+y))f_Y(y)dy$

(assuming X and Y are independent)

**kuywam** · November 23rd 2009, 03:28 PM

Thank you, moo...
Thank you, Laurent...
I will not forget your kind help...^^

**mr fantastic** · November 23rd 2009, 06:38 PM

Originally Posted by kuywam

Thank you, moo...
Thank you, Laurent...
I will not forget your kind help...^^

An acknowledgement (or footnote) in your thesis would be sufficient.

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Thank you so much...^^

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