Does $\mathbb{E}_a[f(x,a)]=0$ have a single root $x^*$ if $f(x,a)=0$ is a single-root function in $x$ for any $a$? Notice that $a$ is random with some given PDF.
2026-04-01 21:58:21.1775080701
Is the expectation of a single-root function, another single-root function?
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No. Let $a$ be a random variable that takes the values $0$ and $1$ with equal probabilities and define $f(x,0)=x$ and $f(x,1)=x^2$. Both functions have the single root $x^*=0$. Then $\mathbb{E}[f(x,a)]=[x+x^2]/2$, which has two real roots, namely $x^*_1=0$ and $x^*_2=-1$.