Let X1,…,Xn be independent Bern(pi ) random variables (i.e., P(Xi =1)=1-P(Xi =0)=pi ). Let Y = X1 + … + Xn.Show that holding E(Y) fixed, Var(Y) is

Let X1,…,Xn be independent Bern(pi ) random variables (i.e., P(Xi =1)=1-P(Xi =0)=pi ). Let Y = X1 + … + Xn.Show that holding E(Y) fixed, Var(Y) is maximized when all the pi are equal. In other words, the variability of the sum is largest when all the summands are alike. Does that match your intuition?