It is assumed that the quality characteristic of interest, denoted by $Y_i$, follows a Normal distribution with mean $μ$ and variance $σ^2$.
We take a sample of size $n$ at time $t$ and measure its quality characteristic. The EWMA statistic at time $t$, denoted by $M_t$, is \begin{align} (Eq.1): M_t = (1-λ)M_{t-1} + λ\bar{Y_t} + k(\bar{Y_t}-\bar{Y_{t-1}}) \end{align} (Eq.1)
where $\bar{Y_t}$ is the sample mean at time t and k is a constant.
Attempt \begin{gather*} E[M_t] = (1-λ)E[M_{t-1}] + λE[\bar{Y_t}] + k(E[\bar{Y_t}]] - E [\bar{Y_{t-1}}]) \end{gather*} Q1: I now assume $E[M_t] = E[M_{t-1}]$, is this correct?
Therefore,
\begin{gather*} E[M_t] = \frac{1}{λ}λμ + k(μ-μ) = μ \end{gather*}
And for the variance,
\begin{gather*} V[M_t] = (1-λ)^2V[M_{t-1}] + λ^2V[\bar{Y_t}] + k^2(V[\bar{Y_t}] + V[\bar{Y_{t-1}}]) + 2(1-λ)(λCov(M_{t-1},\bar{Y_t}) + k( Cov(M_{t-1},\bar{Y_t}) - Cov(M_{t-1},\bar{Y_{t-1}}) )) \end{gather*} Q2: Above, I asumed $\bar{Y_t}$ and $\bar{Y_{t-1}}$ are independent, since $Y_{i,t}$ are independent.
Q3: Is $M_{t-1}$ independent from $\bar{Y_t}$ and $\bar{Y_{t-1}}$?
For my third question, my answer would be that $M_{t-1}$ is dependent on the sample means since there is a relation between them (Eq.1). I don't know if this is trivial but I wanted to make sure I am not missing something here.
Thank you.