Consider a random variable $X$ that can take on the values $0,1$ and $2$. So we have $$p_i = P(X=i), \ i = 0,1,2$$ $$\sum_{i=0}^{2} p_i = 1$$ and $$0 \leq p_i \leq 1$$ So identifying a random variable $X$ is the same thing as identifying a point $(p_0,p_1,p_2) \in \mathbb{R}^3$. Now suppose $X$ is a binomial random variable. Then $$p_i = P(X = i) = \binom{2}{i} q^{i}(1-q)^{2-i}$$ and $$4p_{0}p_{2}-p_{1}^{2} = 0$$
How is this related to algebraic geometry? The above equation is an algebraic variety in $\mathbb{R}^3$?