In certain situations, a random variable $X$ with known mean is simulated to obtain an estimate of $P(X \leq a)$ for some constant given $a$. The simple estimator of a simulation for a run is $I = I (X \leq a)$.
Verify that $I$ and $X$ are negatively correlated.
My approach:
By definition $$Corr(X,I)=\frac{Cov(X,I)}{\sqrt{Var(X)Var(Y)}}$$ Then the only chance for the correlation to be negative is when covariance is negative. $$Cov(X,I)=E[XI]-E[X]E[I]$$
I'm not sure on how to interpret the random variable $I$ and its expected value. Any suggestion would be great!!
Note that $\mathsf{E}I=\mathsf{P}(X\le a)$. When $\mathsf{P}(X\le a)=0$, the covariance between $X$ and $I$ is zero. Otherwise,
\begin{align} \operatorname{Cov}(X,I)&=\mathsf{E}X1\{X\le a\}-\mathsf{E}X\mathsf{P}(X\le a) \\ &= \mathsf{E}[X\mid X\le a]\mathsf{P}(X\le a)-\mathsf{E}X\mathsf{P}(X\le a) \\ &=(\mathsf{E}[X\mid X\le a]-\mathsf{E}X)\mathsf{P}(X\le a)\le 0. \end{align}