# Variance of a white noise process

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.

## 1 Answer

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:

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