Maximum likelihood estimator (Gaussian errors, known SD)

Question

Suppose that the random variables $Y_1, ..., Y_n$, satisfy $Y_i = \beta \cdot x_i + \epsilon_i$ for $i = 1,...,n$ where $\beta$ is a constant, $x_1,...,x_n$, are constants, and $\epsilon_1,...,\epsilon_n$, are independent and identically distributed random variables with $\epsilon_i \sim N(0,\sigma^2)$, where $\sigma^2$ is a known constant.

(a) Determine the exact distribution of $Y_i$.

(b) Find the maximum likelihood estimator $\hat{\beta}$ of $\beta$ and show that it is an unbiased estimator of $\beta$.

(c) Determine the exact distribution of $\hat{\beta}$.

Answer 1

Since $Y$ is the sum of a constant and a normal random variable it has an normal distribution figure out its mean and variance. Write down the likelihood function set the partial derviative of it with respect to $\beta$ to $0$ and solve for $\beta$. Once you have the formula for the estimate of $\beta$ you should be able to figure out its distribution and determine its mean. If the mean turns out to be $\beta$ it is unbiased. I am suggesting to do 3 first and then 2 but it probably can be done either way.