# What is the lower bound of a random variable's variance?

$0$
intuitively $0$
variance using sum square difference is ${\left(x - \mu\right)}^{2}$. There are of course other choices but generally the end result will not be negative. In general the lowest possible value is 0 because if $x = \mu \rightarrow {\left(x - \mu\right)}^{2} = 0$
$x > \mu \rightarrow {\left(x - \mu\right)}^{2} > 0$
$x < \mu \rightarrow {\left(x - \mu\right)}^{2} > 0$