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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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You can think of an improper random variable $X$ as one which has a positive probability of taking on the values $+\infty$ and $-\infty$. It's then not hard to see that $F_X(+\infty) = 1-P(X = +\infty)$, $F_X(-\infty) = P(X = -\infty)$ (where $F(\pm\infty)$ are to be interpreted as limits, as usual).

For instance, if $X_n$ is a constant random variable with $P(X_n = n) = 1$, then as $n \to +\infty$, the $X_n$ are converging to $+\infty$ with probability 1. So the limiting random variable $X$ is improper, and should have $P(X = +\infty) = 1$. The distribution function of $X$ should just be $F_X(t) = 0$, which you may observe is also the (pointwise) limit of the distribution functions $F_{X_n}(t)$.

An improper random variable in this sense never has a finite mean, but the converse is not true. The Cauchy random variable does not have a finite means but is not improper, since the values it takes on are all finite, though they can be very large.

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The key point about improper random variables is their distributions do not satisfy the Kolmogorov probability axioms.

One example might be a distribution where all real values are equally likely, sometimes used as an improper prior distribution in Bayesian statistics.

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I believe the definition in terms of the cumulative distribution function is the right one: $F(\infty) - F(-\infty) < 1$. Improper random variables can be used, for instance, to describe the distribution over time of events that might never happen. For example, the probability $M(t)$ of having been married before age $t$ is a CDF for an improper random variable with PDF $m(t)=M'(t)$, whose integral is less than $1$, because some people never marry.

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I think that's a nice example... –  AIB Jan 19 '11 at 7:44

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