Suppose that $X, Y$ are random variables,both from a probability space to $(0,\infty]$, such that X and $1+Y$ have the same distribution, $Y=Z_1$ with probability equals half, otherwise $Y= \frac{Z_1Z_2} {Z_1+Z_2}$ with probability equals half and $Z_1,Z_2$ are iid and both are copies of $X$.

what can be said about $X$?is $X$ finite almost sure?what about $E(X),Var(X)$?

what i've tried: $E(X)=1+E(Y)=1+\frac{1}{2}E(Z_1)+\frac{1}{2}E(\frac{Z_1Z_2} {Z_1+Z_2})$, because $E(X)=E(Z_1)$,we get $E(X)=2+ E(\frac{Z_1Z_2} {Z_1+Z_2})$.

So $E(X-\frac{Z_1Z_2} {Z_1+Z_2})=2$,it is finite.Therefore $X-\frac{Z_1Z_2} {Z_1+Z_2}$ is almost sure finite.but what about $X$?

Is there any hint how to understand $X$?


  • $\begingroup$ Can you clarify the statement when $Y=Z_1$ or $Y=\frac{Z_1Z_2}{Z_1+Z_2}$ with equal probability ... $\endgroup$ – Liviu Nicolaescu Aug 14 '16 at 14:32
  • $\begingroup$ @LiviuNicolaescu with probability=0.5 we have $Y=Z$ else $Y$ equals the fraction. $\endgroup$ – user115608 Aug 14 '16 at 14:38
  • $\begingroup$ You need to formulate the first paragraph more precisely. As is I cannot make any sense. $\endgroup$ – Liviu Nicolaescu Aug 14 '16 at 17:43
  • $\begingroup$ @LiviuNicolaescu others have understood it.u can see:math.stackexchange.com/questions/1891967/… $\endgroup$ – user115608 Aug 14 '16 at 17:44
  • $\begingroup$ Please formulate the first paragraph precisely This is a research level site and precision is expected in questions. Precision in formulation also increases the chance that you will get a meaningful answer. I am confused by the word when. Does it stand for a conditional if ? Is it a substitute for a logical AND? $\endgroup$ – Liviu Nicolaescu Aug 14 '16 at 17:51

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