# PDF and CDF manipulation

1. May 28, 2009

### S_David

Hello,
I have this equation:

$$\int_{-\infty}^{\gamma}f_X(x)\,dx+\int_{\gamma}^{\infty}F_Y(x-\gamma)\,f_X(x)\,dx$$

where $$f_X(x)$$ and $$F_Y(y)$$ 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:

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

Regards

2. May 28, 2009

Try starting with

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

3. May 28, 2009

### D H

Staff Emeritus
What statdad meant to say was: Try starting with

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

4. May 28, 2009

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

5. May 28, 2009

### S_David

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

6. May 28, 2009

Yes: you can write

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

You should be able to take this and write it as

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

just play around with the integrand.

7. May 28, 2009

### S_David

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:

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

Regards

8. May 28, 2009

Look at the integrand in

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

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.

9. May 28, 2009

### S_David

I can't see anything that I can do. :shy:

10. May 28, 2009

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

11. May 29, 2009

### S_David

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

12. May 29, 2009

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