 Complementary-layer bound for biased measure

Suppose a self-dual set family has level proportions $a_j\le j/n$ for $j<n/2$. Subtract the binomial identity $p={\sum }_j\left(\genfrac{}{}{0ex}{}{n}{j}\right)(j/n)p^j(1-p{)}^{n-j}$ from its biased measure of a set family and pair levels $j,n-j$. The difference isEvery term is nonnegative for $p\ge 1/2$. A self-dual intersecting family has the required level bounds by the Erdős-Ko-Rado theorem.