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Suppose we draw a number $x$ uniformly distributed on $(0,1)$, what is then the following distribution. Furthermore, calculate $F(y)$ and $f(y)$.

$$y = \dfrac{x}{1-x}$$

This is a question I came across. Looks very simple, but I just simply do not know. Especially the functions aren't specified anywhere. Do they have a general meaning?

onimoni
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    $F$ would be the cumulative distribution function of $y$, and $f$ the corresponding density (i.e., $f=F'$). – Harald Hanche-Olsen Mar 24 '14 at 19:26
  • $f(y)$ and $F(y)$ typically denote the probability density and cumulative density function of a random variable. You surely did not see such an underspecified question in any book/forum related to mathematics? – mathse Mar 24 '14 at 19:28
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    @mathse Underspecified? No (but badly formulated, yes). – Did Mar 24 '14 at 19:37

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Note that, for every $y\geqslant0$, $$[Y\leqslant y]=[X\leqslant y/(1+y)], $$ hence $$ F(y)=y/(1+y),\qquad f(y)=\mathbf 1_{y\gt0}/(1+y)^2$$

Did
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  • I have a feeling that this $y$ does not really have a particular name. Is this some known distribution? – onimoni Mar 24 '14 at 19:38
  • @Did. Given your answer, now I can figure out what the question should have been :). The question should have been formulated, in my opinion, as: given that $X$ is a uniformly distributed RV on $[0,1]$, what is the distribution of $Y=\frac{X}{1-X}$. – mathse Mar 24 '14 at 19:51
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    @user61001: wikipedia lists three examples of this type under the category "functions of random variables" http://en.wikipedia.org/wiki/Random_variable#Functions_of_random_variables. Your distribution is not there, though... – mathse Mar 24 '14 at 19:59