Variance of Truncated Normal Distribution

Question

Let $X\sim {N}(\mu,1)$ and let $Y$ denote its truncation for the interval $[0,1]$. What is the relationship between $Var Y$ and $\mu$? Is it monotone? I found that the relationship between $\mathbb{E}Y$ and $\mu$ is increasing with $\frac{\partial \mathbb{E}Y}{\partial \mu}=\frac{Var Y}{Var X}$ and would like to know about $\frac{\partial Var Y}{\partial \mu}$. Thanks in advance!

Answer 1

Use your intuition. The truncated distribution $Y$ will be symmetric when $\mu = 1/2$. As you move $\mu$ away from $1/2$, intuition suggests the variance will not remain constant; therefore, it cannot be monotone--$\mu = 1/2$ must be a global extremum.

The question of whether it is a minimum or maximum will require additional thought.