Joint density of the sum of a random and a non-random variable?

Let's say I have $Y_t = \mu X_t + \sigma Z_t$ with Z IID $N\sim (0,1)$ and $X_t$ as a non-stochastic sequence taking values in the real line. $\mu$ and $\sigma$ are unknown parameters. Eventually I want to find Maximum Likelihood estimators for $\mu$ and $\sigma$ associated with a sample $\{Y_t,X_t\}_{t=1}^n$.

However I am confused how to determine a joint density of a sum of a random $(Z_t)$ and a non random variable $(X_t)$.

What I tried so far is to see X just a bunch of values we observe so we have pairs of $(Z_1, X_1),...,(Z_n, X_n)$. I also solved for Z and assumed the sample $\{Y_t,X_t\}_{t=1}^n$ as

Z = \begin{bmatrix} \frac{Y_1-\mu X_1}{\sigma} \\ . \\ \frac{Y_n-\mu X_n}{\sigma} \end{bmatrix} Since Z is IID with $N\sim (0,1)$ I just used the pdf of standard normal to propose the Maximum Likelihood function

$L(\mu, \sigma^2, \{\frac{Y_t-\mu X_t}{\sigma}\}^n) = \prod_{t=1}^{n}\frac{1}{\sqrt{2\pi}}exp(-\frac{1}{2}(\frac{Y_t-\mu X_t}{\sigma})^2)$

Then I can of course find the maximising values for $\mu$ and $\sigma$.

I am very unsure about this and in particular the step where I solved for Z to obtain the sample feels wrong, since this gives me the joint density $f_Z(z)$ and not $f_Y(y)$ right?

This has a self study purpose so any hints are welcomed.

• How can you know anything about X$_t$?? – Michael Chernick Oct 9 '17 at 23:29
• Hi Michael. $X_t$ is given as a nonstochastic observed sequence in the real line. There is no more information about $X_t$ given. I apologise for being unclear about that. Does that clarify things? – XsLiar Oct 9 '17 at 23:38
• Yes but it gives you 0 information about what specific values might occur and you apparently can't characterize it with a probability distribution. There doesn't seem to be enough information to do what you want. – Michael Chernick Oct 9 '17 at 23:42
• Are the $X$ values known? – Glen_b Oct 10 '17 at 0:47
• Hi Glen. As I stated before the only information that is given about the X values is that they come from a nonstochastic observed sequence in the real line. What do you mean by known? That we assume we know them? – XsLiar Oct 10 '17 at 1:07