## Shifting the mean of simulated Gaussian variables (e.g. using measure change ideas)

I have the following problem (which I actually need to improve a complicated Monte Carlo simulation):

We have n values $X_1, X_2, \dots, X_n$, and we randomly draw n random variables $Y_1, Y_2, \dots, Y_n$, where these variables are independent normally distributed $Y_i \sim N(X_i, \sigma^2)$, were $\sigma$ is a given parameter. Would it be possible to come up with some "weights" $W_{1i}$, such that the random variables $Z_1, \dots, Z_n$, where $Z_i = W_{1i}Y_i$, are independent and normally distributed with variance $\sigma^2$, but all have mean $X_1$?

EDIT: I just realized that the question doesn't really make much sense without additional information:

We are also given values $f(Y_i)$, where $f$ is a deterministic, but otherwise unknown function. I would like to calculate an estimator for $\mathbb{E}[f(Z)]$, where $Z \sim N(X_1, \sigma^2)$. So I am really after weights so that I can write: $$\mathbb{E}[f(Z)] \sim \sum W_{1i} f(Y_i)$$

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