# Question about book solution to estimate $E[e^{XY}]$ when $X$ and $Y$ are independent exponential RVs with $\lambda = 1$

Let $$X$$ and $$Y$$ be independent exponential random variables with mean 1. (a) Explain how we could use simulation to estimate $$E[e^{XY}]$$. (b) Show how to improve the estimation approach in part (a) by using a control variate.

The red box (see screenshot) in the book solution makes sense to me.

I don't understand the blue box. It seems to me the blue box estimator will not have the same expected value as $$E[e^{XY}]$$ and is missing a "$$-c$$" in the numerator because $$E[X_i Y_i] = E[X_i] E[Y_i] = 1$$

So I think the blue box should actually be

$$\sum_{i=1}^n \frac{e^{X_i Y_i} + c X_i Y_i - c}{n}$$

Will the blue box estimator have the same expected value as $$E[e^{XY}]$$? Thanks for your help I'm learning about control variates for the first time.

Book Solution

Yes, for $$c \in \mathbb{R}$$ the first should be $$\frac{1}{n}\sum_{i=1}^n[e^{X_iY_i} + c\underbrace{(X_iY_i-1)}_{\mbox{mean 0}}]$$ and also the second should be $$\frac{1}{n}\sum_{i=1}^n[e^{X_iY_i} + c\underbrace{(X_iY_i + X_i^2Y_i^2/2 - 3)}_{\mbox{mean 0}}]$$ Using $$c=-1$$ seems to be a good choice for reducing variance.