The variance upper bound for a mixed random variable

ABSTRACT

In this note, we derive upper bounds on the variance of a mixed random variable. Our results are an extension of previous results for unimodal and symmetric random variables. The novelty of our findings is that this mixed random variable does not necessarily need to be symmetric and is multimodal. We also characterize the cases when these bounds are optimal.

KEYWORDS:

• Gruss’ inequality
• unimodal
• Popoviciu's inequality
• symmetry
• upper bounds
• variance

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 60E15
• Secondary: 62P05
• 62P20