Is my likelihood function correct? $x_1, ... , x_n$ are known constants and $Y_1, ... , Y_n$ satisfy $Y_i = Bx_i + \epsilon_i$ where $\epsilon_i$ are independent, $N(0, \sigma^2)$, random variables. 
I said my likelihood function is 
$$\prod_{i = 1}^n\frac{1}{\sigma \sqrt{2\pi}} \exp\left({-\frac{(x - Bx_i)^2}{2\sigma^2}}\right)$$
Is this correct
 A: You have already seen that the $Y_i$'s are independent, with $Y_i\sim\mathrm{N}(Bx_i,\sigma^2)$, where the $x_i$'s are known constants. First of all, write down the expression of the densities:
$$
  f_{Y_i}(t) = (?) \, .
$$
That is easy. Just be careful. Now, in your problem you observe the values of the random variables $Y_1,\dots,Y_n$. You need to write the expression of their joint density. Remembering that they are independent, you have
$$
  f_{Y_1,\dots,Y_n}(t_1,\dots,t_n) = f_{Y_1}(t_1) \dots f_{Y_n}(t_n) = (?) \, .
$$
OK, almost there. Here comes the concept of likelihood. Let the observed values of $Y_1,\dots,Y_n$ be $y_1,\dots,y_n$; these little $y_i$'s are your data. The likelihood of $B$ and $\sigma^2$, which you may denote by $L_{y_1,\dots,y_n}(B,\sigma^2)$ is just the expression of $f_{Y_1,\dots,Y_n}(y_1,\dots,y_n)$, which you should already know, seem as a function of $B$ and $\sigma^2$. So, just write it down:
$$
   L_{y_1,\dots,y_n}(B,\sigma^2) = f_{Y_1,\dots,Y_n}(y_1,\dots,y_n) = (?) \, .
$$
