Consider a discrete random variable $X\in[0,1]$ with p.m.f. $f_X(x)$ parametrized by $\theta$. Assume its expected value $\mathbb E[X]$ is increasing in $\theta$.
Is $\mathbb E[X^2]$ increasing in $\theta$ as well?
The equality $\mathbb E[X^2] = \mathbb E[X]^2+Var(X)$ shows that a sufficient condition is for the derivative w.r.t. $\theta$ of $\mathbb E[X]^2+Var(X)$ to be positive. Is there a reason this should be true in general?
Let $X$ take the values $\theta$ and $1-\theta$ with probabilities $\frac 3 4$ and $\frac 1 4$ respectively. You can verify that $EX$ is an increasing function of the parameter $\theta$ on $[0,1]$ but $EX^{2}$ is increasing for $\theta >\frac 1 4$, decreasing for $\theta <\frac 1 4$.