1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Explain this a little

  1. Jul 1, 2007 #1
    Explain this a little plz

    well i have another problem with finding a(some) functions. The problem is this:
    FInd all non-negative continuous functions f:[01]-->R that fullfil the following:

    integ from 0 to 1 of f(x)dx =1

    integ from 0 to 1 of xf(x)dx=a and

    integ from 0 to 1 of x^2 f(x)dx=a^2

    where a is an element of the reals.

    this problem is in a textbook, and it starts like this

    integ from 0 to one of (x-a)^2 f(x) dx, and at the end comes to a contradiction, hence it concludes that such a functions does not exist.

    My question is what is really this problem asking?
    SO if you could explain a little to me how should i start thinking about this problem? Because i want to fully understand what i am writing, and why am i writing something.
    So could anyone please elaborate this problem a little for me???


    I really thank all you guys for your time, and for taking the effort on helping us all.

    P.S. Btw it is not homework, just a problem i encountered in the textbook.
     
    Last edited: Jul 1, 2007
  2. jcsd
  3. Jul 1, 2007 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    Essentially, the int. from 0 to 1 of (x-a)^2f(x) =0, therefore f(x)=0, contradicting the original assumptions.
     
  4. Jul 1, 2007 #3
    Yeah, as i stated i got to this contradiction. I can follow all these steps, but my point is that for example why we even started with the int. from 0 to 1 of (x-a)^2f(x) ? I just can't get the point of this problem, what does it really mean. Because i have done it, but more like in a mechanical way, not a logical one. That is why i want to know the logic behind it, or the reasoning???

    For example, let us suppose that there were no instructions at all, then how would i have to think about this problem in order to start solving it. Because as long as i saw that instruction, i managed to do the rest, but like i stated, not in a reall logical way.
     
    Last edited: Jul 1, 2007
  5. Jul 1, 2007 #4
    so is anyone else going to give me some instructions?
     
  6. Jul 2, 2007 #5
    It's used because it produces a contradiction. It is a contrived question.
     
  7. Jul 4, 2007 #6
    Yeah, i have figured that out as well. Maybe i am being a little ambiguous. What i want to know is that if we were given just the problem

    FInd all non-negative continuous functions f:[01]-->R that fullfil the following:

    integ from 0 to 1 of f(x)dx =1

    integ from 0 to 1 of xf(x)dx=a and

    integ from 0 to 1 of x^2 f(x)dx=a^2

    where a is an element of the reals.

    and not any instruction at all, then how should i tackle this problem????
    Assume that you have never seen this problem solved before, then what would you do?? How would one know at first place that such a function does not exist, or we should just suppose it. But again how to start????
    I want to know how to get the idea of approching this problem, and similar ones?????
     
  8. Jul 4, 2007 #7

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    One wouldn't know such a function did or did not exist. One would try to find what kinds of things these conditions imply. One would think what things it reminds one of. Where do I see integrals of f, xf and x^2f? If I couldn't think of any such cases then I'd just play around - suppose that f has a taylor series and I can do term by term integration, what would that imply? I'd also think about variations on the problem.

    What I personally would notices is:

    it defines is a probability distribution on [0,1] with mean a and second moment a^2, but that implies that the variance is zero. That is clearly impossible, and in fact implies the hint: the variance is the integral of (x-a)^2f(x).



    More abstractly what is really going on is that any such f would have to contradict something like the Cauchy Schwartz inequality.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Explain this a little
  1. Little Proof (Replies: 3)

  2. Little delta (Replies: 0)

  3. Little o function (Replies: 7)

Loading...