Given any discrete random variable $X$, is it always possible to find a change of variables $Y = g(X)$ such that the new variable, $Y$, is uniformly distributed?
$\begingroup$
$\endgroup$
2
-
$\begingroup$ Let $X$ be constant and the probability space $[0, 1]$ with Lebesgue measure. $\endgroup$– AJYCommented Aug 27, 2016 at 17:40
-
1$\begingroup$ It's always impossible. There exists a value $a$ so that $P(X=a)>0$. Hence $P(g(X)=g(a))>0$, so $g(X)$ is not uniformly distributed. $\endgroup$– David C. UllrichCommented Aug 27, 2016 at 17:40
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
If you want $Y$ to be a continuous uniformly distributed variable, then no, that is not possible. Since $X$ is discrete there will be some $x_0$ such that $P(X=x_0)$ is positive, but then $P(Y=g(x_0))$ will be positive too, and a uniform continuous distribution has probability $0$ on every singleton.
You can make $Y$ be a discrete uniform distribution, though you may have to let it be uniform over a single element. Suppose, for example that $X$ is $0$ with probability $\frac23$ and $1$ with probability $\frac 13$. Them your only hope of getting a "uniform" distribution is to let $g(0)=g(1)$.
answered Aug 27, 2016 at 17:42
hmakholm left over Monicahmakholm left over Monica
288k24 gold badges432 silver badges696 bronze badges
-
$\begingroup$ I didn't understand the last part, "a uniform continuous distribution has probability 0 on every singleton". I was asking about a discrete uniform distribution, not a continuous one. Why did you bring that up? $\endgroup$– TenderoCommented Aug 27, 2016 at 18:57
-
$\begingroup$ @Tendero: Bacause I'm not a mind reader. You may have had a discrete uniform distribution in mind, but you didn't say so explicitly in the question. $\endgroup$ Commented Aug 28, 2016 at 10:44