PDF and CDF manipulation

**S_David** · May28-09, 12:25 PM

Hello,
I have this equation:

$LaTeX Code: \\int_{-\\infty}^{\\gamma}f_X(x)\\,dx+\\int_{\\gamma}^{\\infty}F _Y(x-\\gamma)\\,f_X(x)\\,dx$

where

and

are the PDF and CDF of the randome variables X and Y, respectively.

Now the question is: can I write the above equation in the form:

$LaTeX Code: 1-\\int_{0}^{\\infty}(...)$

Regards

**statdad** · May28-09, 02:48 PM

Try starting with

$LaTeX Code: \\int_{-\\infty}^\\gamma f_X(x) \\, dx = \\int_{-\\infy}^\\infty f_X (x) \\, dx - \\infty_\\gamma^\\infty f_X(x) \\, dx = 1 - \\infty_\\gamma^\\infty f_X(x) \\, dx $

**D H** · May28-09, 03:17 PM

What statdad meant to say was: Try starting with

$LaTeX Code: \\int_{-\\infty}^{\\gamma} f_X(x)\\,dx = \\int_{-\\infty}^{\\infty} f_X(x)\\,dx \\;- \\,\\int_{\\gamma}^{\\infty}f_X(x\\,)dx = 1 \\,- \\int_{\\gamma}^{\\infty}f_X(x)dx$

**statdad** · May28-09, 03:42 PM

Originally Posted by D H

What statdad meant to say was: Try starting with

$LaTeX Code: \\int_{-\\infty}^{\\gamma} f_X(x)\\,dx = \\int_{-\\infty}^{\\infty} f_X(x)\\,dx \\;- \\,\\int_{\\gamma}^{\\infty}f_X(x\\,)dx = 1 \\,- \\int_{\\gamma}^{\\infty}f_X(x)dx$

Yes indeed, but statdad, in his advanced age, was interrupted by some annoying folks at the door and neglected to fix his post. Thanks.

**S_David** · May28-09, 03:52 PM

Originally Posted by D H

What statdad meant to say was: Try starting with

$LaTeX Code: \\int_{-\\infty}^{\\gamma} f_X(x)\\,dx = \\int_{-\\infty}^{\\infty} f_X(x)\\,dx \\;- \\,\\int_{\\gamma}^{\\infty}f_X(x\\,)dx = 1 \\,- \\int_{\\gamma}^{\\infty}f_X(x)dx$

Yes, but I want the whole right side be one minus single integral. Is this still doable in some how?

**statdad** · May28-09, 03:56 PM

Yes: you can write

$LaTeX Code: 1 - \\int_\\gamma^\\infty \\left(f_X(x) - F_Y(x-\\gamma)f_X(x)\\right) \\, dx $

You should be able to take this and write it as

$LaTeX Code: 1 - \\int_0^\\infty ( \\cdots ) \\, dx $

just play around with the integrand.

**S_David** · May28-09, 05:47 PM

Originally Posted by statdad

Yes: you can write

$LaTeX Code: 1 - \\int_\\gamma^\\infty \\left(f_X(x) - F_Y(x-\\gamma)f_X(x)\\right) \\, dx $

You should be able to take this and write it as

$LaTeX Code: 1 - \\int_0^\\infty ( \\cdots ) \\, dx $

just play around with the integrand.

Thank you, but this form is not the one in my mind. I need, if possible, in some how, to eliminate the first term, so that the equation looks like:

$LaTeX Code: 1-\\int_0^{\\infty}F_Y(a)\\,f_X(a+\\gamma)\\,da$

Regards

**statdad** · May28-09, 06:07 PM

Look at the integrand in

$LaTeX Code: \\int_\\gamma^\\infty \\left(f_X(x) - F_Y(x-\\gamma)f_X(x)\\right) \\, dx $

You should see a very simple way to factor it and then rewrite it in a form more suitable to your desires for this problem.

Try it - do some work - then post again.

**S_David** · May28-09, 06:19 PM

Originally Posted by statdad

Look at the integrand in

$LaTeX Code: \\int_\\gamma^\\infty \\left(f_X(x) - F_Y(x-\\gamma)f_X(x)\\right) \\, dx $

You should see a very simple way to factor it and then rewrite it in a form more suitable to your desires for this problem.

Try it - do some work - then post again.

I can't see anything that I can do.

**statdad** · May28-09, 07:25 PM

Look a little harder. I won't give away the answer.

**S_David** · May29-09, 08:10 AM

Originally Posted by statdad

Look a little harder. I won't give away the answer.

Just give me a hint, I am not strong in probability.

**statdad** · May29-09, 08:13 AM

Originally Posted by saeddawoud

Just give me a hint, I am not strong in probability.

Work with the integrand.

**S_David** · May29-09, 07:52 PM

Originally Posted by statdad

Work with the integrand.

Are you sure that, we can write $LaTeX Code: \\int_\\gamma^\\infty \\left(f_X(x) - F_Y(x-\\gamma)f_X(x)\\right) \\, dx$ as $LaTeX Code: \\int_0^{\\infty}F_Y(a)\\,f_X(a+\\gamma)\\,da$ ?