I am trying to read through Alon and Frankl's 1985 paper "The maximum number of disjoint pairs in a family of subsets".
Let $\mathcal{F}$ be a family of $m=2^{(1/2+\delta)n}$ subsets of $X=\{1,\dots,n\}$. Let $A_1,\dots,A_t$ be picked independently from $\mathcal{F}$ with repetitions. They prove that if Y is a random variable whose value is the number of members of F that are disjoint to all the $A_i$, then $E[Y]\ge 2m^{1-t\delta^2/2}$.
They then state: "Since $Y\le m$, we conclude that $$\Pr[Y\ge m^{1-t\delta^2/2}]\ge m^{-t\delta^2/2}."$$
How do they conclude this lower bound? I have been trying to use the Paley-Zygmund inequality, but cannot get a handle on $E[Y^2]$.
You can write $$ 2m^{1-t\delta^2/2}\leq E[Y]=E[Y\mathbf{1}\{Y\geq m^{1-t\delta^2/2}\}] + E[Y\mathbf{1}\{Y< m^{1-t\delta^2/2}\}] \\ \leq m P[Y\geq m^{1-t\delta^2/2}] + m^{1-t\delta^2/2}. $$ Now simplify and get $P[Y\geq m^{1-t\delta^2/2}] \geq m^{-t\delta^2/2}$.