View Full Version : finite variance
04-24-2010, 12:20 PM
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?
Thanx in advance for any answers!
04-25-2010, 12:09 AM
What do you think about this?
try one example .. let X follows normal distributon. here the variance is finite
is X bounded?
04-25-2010, 12:36 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.
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