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October 22nd, 2017, 03:35 PM  #1 
Member Joined: Nov 2016 From: Kansas Posts: 73 Thanks: 1  Random Variables
Let X: $\omega$ >R be a realvalued random variable which is not almost surely constant, that is, for every c that belongs to R,we have P [X = c] < 1. Prove: (a) The distribution function of X takes a value in the interval (0, 1). (b) There exist a< b such that 0 < P [X $\leq$ a] < 1 and 0 < P [X $\leq$ b] < 1 
October 22nd, 2017, 05:32 PM  #2 
Senior Member Joined: Aug 2012 Posts: 2,047 Thanks: 585 
Is $\omega$ the natural numbers? Or does it have some other meaning here? I have another question. The phrase "not almost surely constant" is puzzling. By way of context I don't know any probability theory but I know a little measure theory, so I'm trying to translate your terminology to what I know. Now if you tell me a function $f$ is "almost surely constant," that means that it is constant except for a set of measure $0$. For example if $f(x) = 0$ on the irrationals and $f(x) = x$ on the rationals, then $f$ is constant almost everywhere. The set of points where it's not constant, namely the rationals, has measure $0$. In passing this is also problematic for your notation $\omega$, which I know as the settheoretic notation for what's usually called $\mathbb N$, the set of natural numbers. But you can't mean that here, since "almost surely" refers to the set of exceptions having measure $0$ in the domain. But in $\mathbb N$ everything has measure $0$, it's a countable set. So you must mean something else by $\omega$. Now, we've convinced ourselves that a function $f$ is almost surely constant if the set of exceptions (where it's not constant) has measure zero. So if it's NOT almost surely constant, it must be nonconstant on some set of positive measure. For example if the domain is the unit interval, and $f(x) = x$ for $x < \frac{1}{2}$ and $0$ otherwise, then $f$ fails to be constant on a set of positive measure, hence it's "not almost surely constant." I have the feeling this is not what you mean. I can't correspond it to the other characterization that $P [X = c] < 1$. I wonder if you can explain this for my benefit. I'm just trying to expand my own knowledge by understanding your question in terms of what I know. Last edited by Maschke; October 22nd, 2017 at 05:57 PM. 
October 23rd, 2017, 12:49 PM  #3  
Global Moderator Joined: May 2007 Posts: 6,613 Thanks: 617  Quote:
 
October 25th, 2017, 08:46 AM  #4 
Member Joined: Jan 2016 From: Athens, OH Posts: 89 Thanks: 47 
I think this is what you want. If you have questions about the proof or facts stated, just post again. 

Tags 
random, sigma algebra, stochastic process, variables 
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