I'm trying to prove that the variance of a RV whose values are discrete 1's or 0's is greater than the variance of a RV who's values are 0's or continuous on the domain (0,1], where any "1" in the Bernoulli RV corresponds to a value on (0,1] in the other RV. Intuitively, I think this is the case, but I'm trying to demonstrate it.
For instance, I know that $Var[X] > Var[\alpha X]$ where $0 \le \alpha < 1$, but what if $\alpha$ is a RV on the interval $(0,1]$?
The reason I'm asking is that I'm sampling a RV with values on the interval (0,1] and I'm using a binomial confidence interval as an upper bound (you can read more of the context from an old discussion here), and I need to prove that that's a reasonable upper bound on the confidence interval, since the values could all be 1's or 0's, but have the possibility of being between those values.
Edit Sasha provided a counterexample to my question as originally stated. The distribution in question is such that it's Bernoulli-like, however instead of having only 1's and 0's, the "true" values can take on values on the interval (0,1]. So, for Sasha's case of $p_{true} = 0.01$, the distribution in question would have a delta function at 0 of value $1-p_{true}$ and the rest of the distribution would be on (0,1].