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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

enter image description here

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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.

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  • $\begingroup$ Thanks for your help Michael! This was really bothering me, especially as it is the final question of the book that has taken me 2 years of self study to get through lol fml (fantastic book by Ross, I'm just slow) $\endgroup$ – HJ_beginner Nov 17 '18 at 20:32

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