This question involves finding the statistical properties of a discrete-time random process $X[n] = e^{-An}$, where $n \geq 1$ and $A$ is a standard normal random variable (i.e., $A \sim N(0,1)$).

Let's go through each requirement:

1. Expectation Function $E[X[n]]$: To find the expectation of $X[n]$, we calculate $E[e^{-An}]$. Since $A$ is a normal random variable with mean $0$ and variance $1$, we use the moment-generating function of a normal distribution: $E[e^{-An}] = e^{\frac{n^2}{2}}.$

2. Mean Function $E[X[n]]$: The mean function is the same as the expectation function found above, so: [ \text{Mean Function } = E[X[n]] = e^{\frac{n^2}{2}}. ]

3. Variance Function $\text{Var}(X[n])$: To find the variance, we use: [ \text{Var}(X[n]) = E[X[n]^2] - (E[X[n]])^2. ] We calculate $E[X[n]^2] = E[e^{-2An}] = e^{2n^2}$, and thus: $\text{Variance} = e^{2n^2} - (e^{n^2})^2 = e^{2n^2} - e^{2n^2} = 0.$

4. Autocorrelation Function $R_X(n,m) = E[X[n]X[m]]$: The