  Consider a random system $\mathfrak{f}_1(x)=0,\ldots,\mathfrak{f}_n(x)=0$ of
$n$ random real polynomials in $n$ variables, where each $\mathfrak{f}_k$ has a
prescribed set of exponent vectors in a set $A_k\subseteq \mathbb{Z}^n$ of size
$t_k$. Assuming that the coefficients of the $\mathfrak{f}_k$ are independent
Gaussian of any variance, we prove that the expected number of zeros of the
random system in the positive orthant is bounded from above by $4^{-n}
\prod_{k=1}^n t_k(t_k-1)$. This result is a probabilisitc version of
Kushnirenko's conjecture; it provides a bound that only depends on the number
of terms and is independent of their degree.
