How can i understand sth about the distribution of a random variable , if i know that it is identically distributed with a function of it?

I mean if i know $X$ is identically distributed with $f(X)$ , what information can i get about the distribution of $X$?

In particular,I'm thinking about this:

$X$ is identically distributed with $1+Y$ ,when $Y=Z_1$ or $Y= \frac{Z_1Z_2} {Z_1+Z_2}$ with equal probability and $Z_1,Z_2$are iid and both are copies of $X$. How can i know the distribution? Thank you a lot!


closed as unclear what you're asking by Did, Parcly Taxel, heropup, Joey Zou, BruceET Aug 27 '16 at 7:03

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  • $\begingroup$ Any measurable function? continuous? differentiable? injective? bijective? $\endgroup$ – H. H. Rugh Aug 26 '16 at 10:58
  • $\begingroup$ @H.H.Rugh differentiable $\endgroup$ – user115608 Aug 26 '16 at 11:00
  • $\begingroup$ OK, and does $X$ have an $L^1$ density? $\endgroup$ – H. H. Rugh Aug 26 '16 at 11:01
  • $\begingroup$ Which F? The CDF of X? $\endgroup$ – Did Aug 26 '16 at 11:09
  • $\begingroup$ This seems quite open ended, depending on the structure of $F$. For example, if $F(x) = -x$, then any $X$ with a density symmetric about $y$-axis should fall into your category. If $F(x) \equiv a$ for some fixed value $a$ must tell you that $\mathbb{P}(X = a) = 1$. If $F(x) = x^2$, then any Bernoulli random variable should work. If $F$ is invertible, then you can think of when $\mathbb{P}(X \leq t) = \mathbb{P}(X \leq F^{-1}(t))$. $\endgroup$ – Tom Aug 26 '16 at 11:24

Some partial answers: For $A\subset {\Bbb R}$ measurable we should have $$ P_X(A)=P(X\in A) = P(F(X)\in A) = P(X\in F^{-1}A) = P_X(F^{-1}A)$$ Then also $P_X(A)=P_X(F^{-k}A)$ for all $k\geq 1$. So $X$ must have its support in $\cap_k F^k({\Bbb R})$.

If $F$ is $C^1$ and you want to see if $X$ admits a density $f_X\in L^1({\Bbb R})$ then you need $ f_X(y) dy = \sum_{x: F(x)=y} f_X(x) |dx| $ or $$ f_X(y) = \sum_{x:F(x)=y} f_X(x) \frac{1}{|F'(x)|} $$

Something can be said about critical points, and more in the case of expanding maps. But it depends upon your goal.

  • $\begingroup$ i edited the question.note it please.thanks. $\endgroup$ – user115608 Aug 26 '16 at 18:48

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