 My Math Forum Construction of a random variable for a given cdf F
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 April 10th, 2014, 04:01 PM #1 Newbie   Joined: Jan 2014 Posts: 23 Thanks: 0 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. $\displaystyle \lnot \exists x \in \mathbb{R} : F(x) = e$? Tags cdf, construction, random, variable Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Keroro Advanced Statistics 2 August 17th, 2012 01:22 PM lawochekel Algebra 1 April 19th, 2012 12:39 PM varunnayudu Advanced Statistics 1 November 28th, 2010 06:33 AM meph1st0pheles Advanced Statistics 0 February 27th, 2010 02:03 PM xiongzi Advanced Statistics 0 March 12th, 2009 06:24 AM

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