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Probability density functions

  1. Nov 9, 2007 #1
    1. The problem statement, all variables and given/known data

    3. The attempt at a solution
    I know how to compute something like Pr(x<0.25) for example, but I'm unsure how to do it for an exact number like in question (ii). I attempted to integrate and then sub x=1/4 where neccisary, but to no avail!

    Part (iii) I really have no idea!

    Would be grateful is someone could explain these to me. Thanks! :cool:
  2. jcsd
  3. Nov 9, 2007 #2

    D H

    Staff: Mentor

    ii) Is a trick question in a sense. Hint: the P(X=1/4) is the same for all continuous random variables with a continuous pdf.

    iii) What is the definition of expected value?
  4. Nov 9, 2007 #3
    Ok, so P(x=1/4)=0 for all condinuous random variables with a continuous pdf, how come?

    Re part (iii) straight swap (cos(pi*x)) for X into this equation?
    If I do that, i get a nasty integral of cos(pi*x)*sin(pi*x)dx

    Pardon my silly questions, I'm rusty as hell..
    Last edited: Nov 9, 2007
  5. Nov 9, 2007 #4

    D H

    Staff: Mentor

    For any continuous function f, what is

    [tex]\lim_{\epsilon\to 0}\int_{a-\epsilon}^{a+\epsilon}f(x)dx[/tex]

    Re part (iii). This is a simple integral. If the factor of pi is throwing you off, think of this as

    [tex]\int \cos(ax)\sin(ax)dx[/tex]

    and then set [itex]a=\pi[/itex] after integrating.
  6. Nov 9, 2007 #5
    Ah ha, got you for part (ii), it's not just 1/4 then, it's any number correct!?

    re (iii), i'm still bloody lost. tried integration by parts, but the integration part just returns the same integral so it's like going around in a circle (if you know what i mean). I can't think of any other way to solve it :confused:
  7. Nov 9, 2007 #6

    D H

    Staff: Mentor

    Try a simple u-substitution.
  8. Nov 10, 2007 #7


    User Avatar
    Staff Emeritus
    Science Advisor

    Let u= sin([itex]\pi[/itex]x).
  9. Nov 10, 2007 #8
    Ah gotcha!
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