Nilotpal Sinha
Reputation
1,388
Next privilege 2,000 Rep.
 2d awarded Yearling Feb 25 awarded Popular Question Jan 30 awarded Nice Question Jan 7 awarded Nice Answer Nov 29 awarded Popular Question Oct 22 awarded Nice Answer May 15 revised Why could Mertens not prove the prime number theorem? [Edit removed during grace period] May 2 awarded Yearling Mar 31 awarded Notable Question Mar 6 awarded Popular Question Sep 25 awarded Favorite Question Jul 2 awarded Curious May 12 awarded Notable Question May 2 awarded Yearling Aug 16 revised Is there a probability density function satisfying the following conditions? edited tags Aug 15 comment Is there a probability density function satisfying the following conditions? @Fedja: No I still don't understand. Can you exemplify with one working example of a solution that works for all $y > 0$. Aug 15 comment Is there a probability density function satisfying the following conditions? @Nik, I had the same formulation that you have given and I can draw several distributions through $(0,1)$ and $(1,e^{-y})$. However satisfying condition 5 is the problem. For example $p(x,y) = e^{-yx^a}$ satisfies the first four conditions. Now if conditions 5 is also going to be true the we find that $y$ must be exactly equal to $\Gamma(1+1/a)^a$. But the minimum value of $\Gamma(1+1/a)^a$ is $e^{-\gamma}$. So this solutions is good for $y > e^{-\gamma}$ but it does not hold for $0 < y < e^{-\gamma}$. Likewise I have not been able to find a function $p(x,y)$ which satisfies 5 for all $y > 0$. Aug 15 awarded Necromancer Aug 15 awarded Nice Question Aug 14 comment Is there a probability density function satisfying the following conditions? @NikWeaver: Yes that is correct