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April 10th, 2014, 04:01 PM   #1
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Construction of a random variable for a given cdf F

Recently I've read a proof of the fact that every function $\displaystyle F$ that is nondecreasing, left-continuous with limits $\displaystyle F(-\infty) = 0$ and $\displaystyle F(+\infty) = 1$ is a CDF of a certain random variable (I know, that in most textbooks a CDF is assumed to be right-continuous, but in this one they use left-continuity - it does not change the problem much).

The proof proceeds by taking the Lebesgue measure on $\displaystyle [0, 1]$ as a probability measure and then the random variable is defined as $\displaystyle X(e) = inf \{ y : F(y) = e \}$. I think, however, that this approach is wrong, because if the function $\displaystyle F$ has a "jump" at a certain $\displaystyle x_0$, let's say from $\displaystyle e_1$ to $\displaystyle e_2$ then for any $\displaystyle e \in (e_1, e_2)$ the value of the random variable $\displaystyle X$ is $\displaystyle +\infty$. Why not just take $\displaystyle X(e) = inf \{ y : F(y) \geq e \}$ ? I think it should work fine.

A little more explanations of my point of view: since the task is to construct a random variable $\displaystyle X$ that has a cdf described by a given function $\displaystyle F$ then, obviously, we cannot assume that the variable $\displaystyle X$ that we just constructed has a cdf $\displaystyle F$, this has to be achieved by properly constructing $\displaystyle X$ and then proving that its cdf is $\displaystyle F$. The idea of the author of the book was to define the random variable $\displaystyle X$ as $\displaystyle \forall e \in \left[0, 1\right] : X(e) = inf \{ y : F(y) = e \}$ for a given function $\displaystyle F$. According to the author of the book the variable $\displaystyle X$ should then have the cdf $\displaystyle F$. IMHO this approach fails, because if you take any value $\displaystyle e$ for which $\displaystyle \lnot \exists x \in \mathbb{R} : F(x) = e$ then we get $\displaystyle X(e) = +\infty$. My suggestion is to define X as $\displaystyle X(e) = inf \{ y : F(y) \geq e \}$.

I would just like to verify if the construction with $\displaystyle X(e) = inf \{ y : F(y) \geq e \}$ is correct and whether the construction with $\displaystyle X(e) = inf \{ y : F(y) = e \}$ has the problem I mentioned above. Maybe I'm wrong and the definition given by the author of the book works, but then what is the value of the random variable $\displaystyle X(e)$ if we choose $\displaystyle e$ s. t. $\displaystyle \lnot \exists x \in \mathbb{R} : F(x) = e$?
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