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leo nidas
04-24-2010, 01:20 PM
Hi there,

if X is a random variable and all we know is that it has finite variance. Can we say that X is bounded? If not what is the meaning of the restriction of a finite variance?

vinux
04-25-2010, 01:09 AM

try one example .. let X follows normal distributon. here the variance is finite

is X bounded?

statsguy
04-25-2010, 01:36 AM
The answer is no. We cannot determine boundedness of a RV X based on knowing only the variance of X. Two simple examples:

Suppose we know that X has finite variance.

Let X ~ N(0,1). Then P(X > c) > 0 for any real number c even though \sigma^2 = 1. That is, no matter how large a c we choose, there is a nonzero probability (p > 0) that X will be greater than c. So there exists no M such that P(X < M) = 1 a.e. So X is unbounded.

Let X ~ U(0,1). X is bounded.

In the reverse direction:

A RV that is bounded will have finite variance. This can be easily proved. A random variable that is unbounded may have finite or infinite variance.