For a given time series \(x_{1}, x_{2}, \ldots\), the \(N\)-period arithmetic Moving Average (MA), \(A(n, N)\), is defined as

\[A(n, N)= \begin{cases}\left(x_{1}+\ldots+x_{n}ight) / n & \text { if } n

Similarly, given \(0<\alpha<1\), the Exponential Moving Average (EMA), \(E(n, \alpha)\), is defined as

\[E(n, \alpha)= \begin{cases}x_{1} & n=1 \\ \alpha x_{n}+(1-\alpha) E(n-1, \alpha) & n>1\end{cases}\]

When computing \(A(n, N), E(n, \alpha)\), the age of a data point \(x_{n-i}\) is \(i\), and the Average Age is the weighted average age.

(a) Compute the average age of an \(\mathrm{N}\)-period arithmetic MA.

(b) Prove the following

\[E(n, \alpha)=\sum_{i=0}^{n-2} \alpha(1-\alpha)^{i} x_{n-i}+(1-\alpha)^{n-1} x_{1}\]
and verify the following asymptotic approximation for large \(n\)
\[E(n, \alpha) \approx \sum_{i=0}^{\infty} \alpha(1-\alpha)^{i} x_{n-i}\]

(c) Using the approximation from part (b), verify that the weights \(\alpha(1-\alpha)^{i}\) add up to 1 , and compute the asymptotic Average Age of EMA \[\sum_{i=0}^{\infty} i \alpha(1-\alpha)^{i}\]

(d) An EMA with an asymptotic average age equal to an \(\mathrm{N}\)-period arithmetic MA is called an \(\mathrm{N}\)-period EMA. What is the \(\alpha\) of an \(\mathrm{N}\)-period EMA?

(e) A crypto exchange defines its settlement price as the 30-second EMA of the Bitcoin price. What weight is assigned to the Bitcoin price of 15 seconds ago?