Is it possible to find two real $X, Y$ IID random variables with different density function ?
If we denote $f_X$ the density function of $X$ and $f_Y$ the density function of $Y$ it would mean that for all $[a,b] \subset \mathbb{R}$ we have :
$$ \int_a^b f_X(x) \mathrm{d}x = \int_a^b f_Y(x) \mathrm{d}x$$ ThusI am wondering if it's possible ?
Thank you !
Every random variable admits a cumulative distribution function (c.d.f). Now if those c.d.f's are absolutely-continuous which is the "common" case, then the must have the same densities1. Then the c.d.f's must be the same because:
Your r.h.s is $~~~F_X(b)-F_X(a)~~~$ and your l.h.s is $~~~F_Y(b)-F_Y(a)$.
Those have to be equal $\forall a,b$ pairs. Picking $a\to -\infty$ we have:
$F_X(b)=F_Y(b) ~~~ \forall b$
Thus having the same absolutely-continuous cdf means that they have the same densities. In the case of non-absolutely continuous cdf's I think it becomes a lot trickier.
1possibly different in null sets