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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!

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

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ 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
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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.

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

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