Is $\mathrm{Var}(Y)\leq \max \mathrm{Var}(Y|X)$?

Question

Let $E_1,E_2,\cdots, E_n$ be mutually exclusive and disjoint events on a probability space. Let $Y$ be a random variable on the same space.

I am trying to check if $\mathrm{Var}(Y)\leq \max_i \mathrm{Var}(Y|E_i)\tag{1}$

By Law of total variance $\mathrm{Var}(Y)=\mathbb{E}[\mathrm{Var}(Y|X)]+\mathrm{Var}(\mathbb{E}[Y|X])$. So the first term in the total variance formula is less than or equal to $\max_i \mathrm{Var}(Y|E_i)$ but may I know how much can the second term contribute? Is there an example where $(1)$ is false?

Answer 1

Let the probability space have $n$ points $p_1,..., p_n$, let $Y$ be injective, and let $E_i$ be the event $Y=Y(p_i)$.