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What is an Improper Random Variable? I know the definition in terms of the CDF like, F(∞) - F(-∞) <1. Could any one explain it more clearly, specifically I am looking for an example of an improper random variable. Is Cauchy distributed Random Variable improper? (Since it has a long tail)

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closed as off topic by Qiaochu Yuan, Willie Wong, Bill Johnson, Nate Eldredge, Pete L. Clark Jan 19 '11 at 8:29

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This question is not really appropriate for MO; you should ask on one of the other sites in the FAQ, such as – Qiaochu Yuan Jan 18 '11 at 14:19
Should I delete the post here and ask in the site you mentioned ? – AIB Jan 18 '11 at 14:24
Since no one has answered, deleting the question is reasonable. I see that someone has already answered it at math stack exchange. – arsmath Jan 18 '11 at 15:27
up vote 1 down vote accepted

Consider a US quarter. Define a random variable to be 1 if the coin lands "heads", 0 if it lands tails. This variable is improper, since there is a chance, albeit small, that the coin will land on edge.

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This is a cute illustration, provided one has an inkling what the definition/description of an improper random variable actually is. – Willie Wong Jan 18 '11 at 18:17
@Igor: I disagree. The random variable you describe is either not improper (if the chance to land on edge is null) or not defined (if it is positive). To get an improper random variable, one must define it as you did AND further define it to be infinite when the coin lands on edge AND assume that this happens with positive probability. – Did Jan 18 '11 at 19:43
You guys are talking about something different from what I wrote about in my answer. I cited a book. Is there a source for the concept you're talking about? Do you just mean a random variable that can assume infinite values with positive probability? – Michael Hardy Jan 18 '11 at 20:22
That's what the question is about: assuming infinite values with positive probability. It's not about improper priors. The CDFs of an improper prior wouldn't satisfy F(\infty) - F(-\infty) < 1 (an improper prior doesn't even have a CDF in the usual sense.) – arsmath Jan 18 '11 at 20:31
@Michael: Yes. See the answer on math.stackexchange. – Did Jan 18 '11 at 20:31

The term "improper prior" is commonplace. "Improper probability distribution" may occur. It's a non-negative-valued measure that assigns infinite measure to a whole space, and thus is not a probability distribution, but that gets treated in some contexts as if it were a probability distribution. For example, suppose the distribution of $m$ to be Lebesgue measure on the line (which of course is not a probability distribution), and that the conditional distribution of $X_1,\dots,X_n$ given $m$ is that they are independent and normally distributed with expectation $m$ and variance $1$. Then ask what is the conditional probability distribution of the unobserved $m$ given the observed values of the $X$s. You get a "proper" probability distribution---the "posterior" distribution of $m$. The "prior" distribution of $m$ in this case is improper.

This makes one's inferences about $m$ translation-equivariant.

Added in a later edit: Harold Jeffreys' book Theory of Probability may be the source of the fact that this concept became standard. Another person who argued on many occasions in favor of the use of improper priors was the physicist Edwin Jaynes.

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Wikipedia gives these 3 as examples of improper priors 1. Beta(0,0), the beta distribution for α = 0,β = 0. 2. The uniform distribution on an infinite interval (i.e., a half-line or the entire real line). 3. The logarithmic prior on the positive reals. The 2nd one is clearly improper by observation...Is improper prior same as improper random variable? – AIB Jan 19 '11 at 11:48
Apparently "improper random variable" does not mean the same thing as a random variable whose distribution is improper, if we can believe the comments accompanying the other answer. – Michael Hardy Jan 19 '11 at 20:31
Michael: There is no such thing as a random variable whose distribution is improper if one gives to improper its Bayesian meaning in the expression improper prior, i.e. of infinite total mass. This is because there is no way to assign to a random variable $X$ a measure of mass $> 1$ which one could call its distribution. On the contrary, when $X$ is infinite with positive probability, the total mass of the restriction of $P_X$ to $\mathbb{R}$ is $< 1$ (and $P_X$ itself is sometimes called defective). – Did Jan 21 '11 at 12:17
If you take the terms literally, there may be no such thing as a random variable whose distribution is improper, but any time you assign an improper distribution to an unobservable parameter, that parameter could plausibly be spoken of as a random variable whose distribution is improper. Of course, someone like Jaynes might object that one should never use the word random in probability theory, but few people are purists like him. – Michael Hardy Jan 22 '11 at 23:29
I see your point and I disagree with it (even though I use the word random routinely). First, I know of no setting where the prior distribution for a given parameter is improper and the parameter is spoken of as a random variable. Could you name some examples where this is actually done? Second, to speak of random variables in this situation soon becomes slippery. (to be cont'd) – Did Jan 23 '11 at 16:27

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