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How do I find the variance of this? $$y_t=M + \sum_{i=0}^n 0.5^i\varepsilon_{t-2i}$$ M is the mean.

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Knowing tht the variance of the sum of independent random variables is the sum of their variances and that $Var(a+bX) = b^2Var(X)$, where $a$ and $b$ are constant:

$$Var(y_t) = \sum_{i=0}^n0.5^{2i}Var(\epsilon_{t-2i})$$

You can solve the sum of the geometric series by using:

$$\sum_{k=0}^{n-1}ar^k = a \frac{1-r^n}{1-r}$$

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