# 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

• It cannot be correct because it includes an undefined symbol "$x$". While you're pondering that, double-check the pdf for the standard normal distribution and notice the factor of $1/2$ in the exponent--that mustn't be overlooked, either. – whuber Feb 13 '13 at 21:03
• You need some notation for the data: notice that the $x_i$ are not data. Start with a dataset of one value and write down its likelihood in terms of the data value, $B$, $\sigma$, and $x_1$. The rest is easy, because independence of the data implies the likelihood when $n\gt 1$ is the product of the likelihoods of the data. – whuber Feb 13 '13 at 21:11
• You need names for each of the numbers in the dataset whose likelihood you wish to write down. You haven't even mentioned those data yet, which is why you're struggling. (The $Y_i$ are random variables, not data values.) – whuber Feb 13 '13 at 21:13
• Yes, you have provided all the information needed to write a likelihood (assuming independent data), because you have specified the distributions. But my writing it down won't help you achieve the understanding you seek: obviously you have seen likelihood expressions before, but now you're having to work out some details that weren't apparent to you when you read those expressions. The best I can do for you is to help you focus your energies so you don't waste effort. – whuber Feb 13 '13 at 21:20
• @whuber: I tried to help without spoiling everything. – Zen Feb 14 '13 at 4:05

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: $$L_{y_1,\dots,y_n}(B,\sigma^2) = f_{Y_1,\dots,Y_n}(y_1,\dots,y_n) = (?) \, .$$