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Proving function f(x) is a PDf given integral relationships

  1. Oct 13, 2016 #1
    1. The problem statement, all variables and given/known data
    Prove that ##\int _0^a\left(\frac{f\left((sin(ln(c))x\right)+\sqrt{\cos \left(e-\pi ^2\right)}}{\ln \left(\pi ^2-e\right)+\pi ^2\sqrt{\cos \left(e-\pi ^2\right)}}\right)dx## is a probability density function (when ##a=\frac{1}{\pi ^2}##) given that ##\int _0^{\pi ^2}f\left(x\right)dx=\ln \left(\pi ^2-e\right)## and that ##c## is a maximum.

    2. Relevant equations
    ##\int _a^bf\left(x\right)dx=F\left(b\right)-F\left(a\right)##

    3. The attempt at a solution
    I'm honestly completely stuck with this question. I know that ##\int _0^a\left(\frac{f\left((sin(ln(c))x\right)+\sqrt{\cos \left(e-\pi ^2\right)}}{\ln \left(\pi ^2-e\right)+\pi ^2\sqrt{\cos \left(e-\pi ^2\right)}}\right)dx## can be expressed as ##\frac{1}{\ln \left(\pi ^2-e\right)+\pi ^2\sqrt{\cos \left(e-\pi ^2\right)}}\int _0^a\left(f\left((sin(ln(c))x\right)+\sqrt{\cos \left(e-\pi ^2\right)}\right)dx##, but I am unsure of how this helps me to prove that the integral is equal to 1 (thus proving it is a PDf). By specifying that ##c## is a maximum, the question insinuates that ##c## is a variable that can be expressed in terms of another variable which in turn can be derived and solved, however I am unsure of how to form such a relationship.
     
  2. jcsd
  3. Oct 13, 2016 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    c is a maximum of what?

    There are mismatched brackets related to f.
    It looks like the problem depends on c, which is odd. What exactly do you know about c?
     
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