Finding the Probability Density Function of Y with Given Continuous PDF f(x)

[dkotschessaa]

Homework Statement

Let the random variable X have the continuous probability density function f(x). find the probability density function of Y= F(x)

Homework Equations

I *think* we are meant to use something like $h(y) = f(g^{-1}(y)) * ((dg^{-1}(y)) /dy)$

(That is as good as I know how to format right now - I hope that's readable).


But I'm given just f(x) and I'm being asked for F(x). And yet it's not asking for a CDF, it's asking for a PDF. So I'm really confused on what's being asked for here.

(Unless this is another one of my book typos) Does the question even make sense?

-Dave K

 

[Ray Vickson]

Homework Statement

Let the random variable X have the continuous probability density function f(x). find the probability density function of Y= F(x)

Homework Equations

I *think* we are meant to use something like $h(y) = f(g^{-1}(y)) * ((dg^{-1}(y)) /dy)$

(That is as good as I know how to format right now - I hope that's readable).


But I'm given just f(x) and I'm being asked for F(x). And yet it's not asking for a CDF, it's asking for a PDF. So I'm really confused on what's being asked for here.

(Unless this is another one of my book typos) Does the question even make sense?

-Dave K

Find the cdf of Y as $F_Y(y) = P\{ Y \leq y \} = P\{ F(X) \leq y \},$ then find the density of Y (the pdf) by differentiation. This method applies even when the formula you use fails (for example, because of intervals or points where f(x) = 0 but inside the range of X). Doing a bit of on-line research, or looking into a book on simulation, may be very helpful.

 

[dkotschessaa]

Find the cdf of Y as $F_Y(y) = P\{ Y \leq y \} = P\{ F(X) \leq y \},$ then find the density of Y (the pdf) by differentiation. This method applies even when the formula you use fails (for example, because of intervals or points where f(x) = 0 but inside the range of X). Doing a bit of on-line research, or looking into a book on simulation, may be very helpful.

So am I to integrate first for the CDF, and then differentiate back down again?

I think it's perhaps so trivial that I'm getting confused. I even have the "back of book" answer but I don't know the reasoning.

 

[Ray Vickson]

So am I to integrate first for the CDF, and then differentiate back down again?

I think it's perhaps so trivial that I'm getting confused. I even have the "back of book" answer but I don't know the reasoning.

That is why I suggested you do a bit of on-line research or look into a book on simulation. Specifically, look at 'methods of generating random samples' or 'random variable generation' or something similar.

 

[dkotschessaa]

That is why I suggested you do a bit of on-line research or look into a book on simulation. Specifically, look at 'methods of generating random samples' or 'random variable generation' or something similar.

I've looked everywhere online. But I'll try that.

 

[vela]

Is Y=F(x) exactly how it was written in your book? If so, it looks like a typo or two. You may want to ask your professor for clarification.

Perhaps you're supposed to find the cdf F(x) and then define the new random variable Y=F(X) (a function of X, not x) and find its density function.

 

[dkotschessaa]

Is Y=F(x) exactly how it was written in your book? If so, it looks like a typo or two. You may want to ask your professor for clarification.

Yes, and the book is full of typos. And the professor gets lost in the middle of class. and..and.. you get the idea.

Perhaps you're supposed to find the cdf F(x) and then define the new random variable Y=F(X) (a function of X, not x) and find its density function.

I'm not sure. I'm really confused and frustrated. I have something written down and I'm handing it in.

Thanks for the help everyone.

-Dave K

 

[ArcanaNoir]

Dang David, next time try running it by me. I already had that class so I should be able to work it out if I can see the problem in the book. :)

 

[haruspex]

Is Y=F(x) exactly how it was written in your book? If so, it looks like a typo or two. You may want to ask your professor for clarification.

Perhaps you're supposed to find the cdf F(x) and then define the new random variable Y=F(X) (a function of X, not x) and find its density function.

No, I think Ray's interpretation is right. Yes, it should be Y=F(X), not F(x), but I doubt there's any relationship between f and F.

 

[vela]

But it doesn't sound like F(X) was specified, which makes it impossible to find f_Y(y).

 

[Ray Vickson]

But it doesn't sound like F(X) was specified, which makes it impossible to find f_Y(y).

No. Finding $f_Y(y)$ is "easy" (once you see it!); this is one of the important standard results that are used all the time in Monte-Carlo simulation.

 

[vela]

I don't see it. You're given random variable X with probability density f_X(x). Then you're asked to find f_Y(y) where Y=F(X). Haruspex has suggested F(X) bears no relation to f_X(x), so it's completely arbitrary. You have virtually no information about Y other than it's a function of X.

 

[haruspex]

I don't see it. You're given random variable X with probability density f_X(x). Then you're asked to find f_Y(y) where Y=F(X). Haruspex has suggested F(X) bears no relation to f_X(x), so it's completely arbitrary. You have virtually no information about Y other than it's a function of X.

The answer will involve F'.

 

[Ray Vickson]

I don't see it. You're given random variable X with probability density f_X(x). Then you're asked to find f_Y(y) where Y=F(X). Haruspex has suggested F(X) bears no relation to f_X(x), so it's completely arbitrary. You have virtually no information about Y other than it's a function of X.

I will be happy to elucidate as soon as I am convinced the OP has already handed in his solution.

 

[Ray Vickson]

Homework Statement

Let the random variable X have the continuous probability density function f(x). find the probability density function of Y= F(x)

Homework Equations

I *think* we are meant to use something like $h(y) = f(g^{-1}(y)) * ((dg^{-1}(y)) /dy)$

(That is as good as I know how to format right now - I hope that's readable).


But I'm given just f(x) and I'm being asked for F(x). And yet it's not asking for a CDF, it's asking for a PDF. So I'm really confused on what's being asked for here.

(Unless this is another one of my book typos) Does the question even make sense?

-Dave K

In my previous responses I have been assuming all along that F is the cdf of X (just because of the notation you have used). If that is not what you meant then all my previous responses are irrelevant.

 

[ArcanaNoir]

I will be happy to elucidate as soon as I am convinced the OP has already handed in his solution.

I can attest to the fact that the assignment has been handed in. OP is on vacation now so you probably won't hear from him for a week.

 

[Ray Vickson]

I can attest to the fact that the assignment has been handed in. OP is on vacation now so you probably won't hear from him for a week.

OK, so if X has pdf f(x) and cdf F(x), then Y = F(X) is uniformly distributed on (0,1). This is true even if X does not have a density (such as for discrete random variables, or mixed random variables that are partly continuous and partly discrete); the only thing that matters is that F(x) is a well-defined function (defined so as to be right-continuous). See, eg.,
http://en.wikipedia.org/wiki/Inverse_transform_sampling
for illustration and proof.

This method is of fundamental importance in Monte-Carlo simulation.

Note: the OP did not actually *say* that F was the cdf of X, but I just assumed he meant that. If not, then F is a more-or-less arbitrary function and there will be no simple, general result.

 

[dkotschessaa]

Dang David, next time try running it by me. I already had that class so I should be able to work it out if I can see the problem in the book. :)

Oh, the whole assignment was a mess, as you know. I'd be bugging you for every question.

I am dropping the class anyway. My first drop, ever. I took some time on spring break to think about it. (I'm still there, just at a hotel in the middle of nowhere, PA). My final act was to get together with the class (we are down to about 8 people) and we agreed on some solutions to the assignment.

Thanks everyone.

-Dave K

 

[HallsofIvy]

In post 17, Ray Vickson is assuming that F(x) is the integral of f(x). Is that true? My first thought was that they were unrelated functions.

In that case, $$P(y< Y)= P(F(x)< Y= \int_{-\infty}^Y F(t)f(t)dt$$ and the PDF is the derivative of that, with respect to Y.

 

[dkotschessaa]

The question was word for word as found in the book. There was no further information and I don't know what we were supposed to assume.

 

[Ray Vickson]

The question was word for word as found in the book. There was no further information and I don't know what we were supposed to assume.

Some (not all) books adopt a convention such as using the upper-case letter for the cdf corresponding to the lower case letter for the pdf, so f is paired with F, g is paired with G, etc. Maybe your book says somewhere that it will do that; I don't know, but that is a pretty common convention for writers of probability textbooks.