Bounded Probability Density Function

[robbins]

Let the random variable X have the probability density function f(x). Suppose f(x) is
continuous over its domain and Var[X] is bounded away from zero: 0 < c < Var[X].

Claim: f(x) is bounded over its domain.

Is this claim true?

I don't think a counterexample like X ~ ChiSq_1 applies because, while f(0) is not bounded, the domain is x > 0. The question of continuity at 0 doesn't arise since 0 is not part of the domain, and therefore the lack of boundedness at 0 isn't relevant. But for any given c > 0, I can find an M such that f(c) < M. This example does show, however, why one can't claim "uniformly bounded."

Similarly, a degenerate r.v. (all probability mass at the point X = c, say) is continuous (because it is discrete), but the variance is not bounded away from zero.

So the question remains: is the claim true? How might one prove it?

Thanks.

 

[quadraphonics]

Similarly, a degenerate r.v. (all probability mass at the point X = c, say) is continuous (because it is discrete), but the variance is not bounded away from zero.

That example isn't relevant, because a degenerate r.v. does not have a p.d.f.

 

[statdad]

Let the random variable X have the probability density function f(x). Suppose f(x) is
continuous over its domain and Var[X] is bounded away from zero: 0 < c < Var[X].

Claim: f(x) is bounded over its domain.

Is this claim true?
No, it is not, and the \chi^2_1 distribution you mention shows it. You do not need to worry about the pdf at x = 0, since the domain is only (0, \infty). The variance of \chi^2_k is \sigma^2 = 2k, so the variance of \chi^2_1 is bounded away from zero. However, the density is not bounded on its domain.
Statdad