MHB Finding Marginal PDFs: Need Help With Integrating Complex Expressions?

  • Thread starter Thread starter nacho-man
  • Start date Start date
  • Tags Tags
    Marginal
nacho-man
Messages
166
Reaction score
0
Please see the attached image for my question. I don't understand how to compute the integral, is there some trick?
I do believe that
to find fx(x) I integrate the joint pdf, with respect to x with the bounds set as the range of Y. But this leaves me with a very complex integration
Similarly for fy(y). Is there some trick?

Any help is greatly appreciated.
 

Attachments

  • pdf q.jpg
    pdf q.jpg
    23.7 KB · Views: 95
Last edited by a moderator:
Physics news on Phys.org
nacho said:
Please see the attached image for my question. I don't understand how to compute the integral, is there some trick?
I do believe that
to find fx(x) I integrate the joint pdf, with respect to x with the bounds set as the range of Y. But this leaves me with a very complex integration
Similarly for fy(y). Is there some trick?

Any help is greatly appreciated.
Hi nacho!

To find $f_X(x)$ you need to integrate $f_{X,Y}(x,y)$ with respect to y.

That is:
$$f_X(x) = \int_{-\infty}^{+\infty} f_{X,Y}(x,y) dy$$

The problem is of course that exponent, but you can complete the square, do a substitution, and use the standard integral of a normal distribution.

That is:
$$f_X(x) = \int_{-\infty}^{+\infty} f_{X,Y}(x,y) dy
= \int_{-\infty}^{+\infty} \frac{1}{\pi\sqrt 2} \exp\left(-(y + \frac 1 2 x \sqrt 2)^2 - \frac 1 2 x^2\right) dy$$

Can you substitute $w = y + \frac 1 2 x \sqrt 2$?

And use that $$\int_{-\infty}^{+\infty} \exp\left(-\frac 1 2 u^2\right) du = \sqrt{2\pi}$$?
 
Hi all, I've been a roulette player for more than 10 years (although I took time off here and there) and it's only now that I'm trying to understand the physics of the game. Basically my strategy in roulette is to divide the wheel roughly into two halves (let's call them A and B). My theory is that in roulette there will invariably be variance. In other words, if A comes up 5 times in a row, B will be due to come up soon. However I have been proven wrong many times, and I have seen some...
Thread 'Detail of Diagonalization Lemma'
The following is more or less taken from page 6 of C. Smorynski's "Self-Reference and Modal Logic". (Springer, 1985) (I couldn't get raised brackets to indicate codification (Gödel numbering), so I use a box. The overline is assigning a name. The detail I would like clarification on is in the second step in the last line, where we have an m-overlined, and we substitute the expression for m. Are we saying that the name of a coded term is the same as the coded term? Thanks in advance.

Similar threads

Back
Top