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)
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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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.
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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