How to represent normal law and a variable intermingled?

Question

Let's consider that a variable y constructed from x

$x_i ∈ \left\{1;3;5;7;8\right\}$

$f(x_i)=2x_i+1$

$y_i=f(x_i) + ε_i, ∀i∈ \left\{1;...;5\right\} $

where $ε_i$ is a identically and independantly distributed random variable which follows a normal law $\mathcal{N(0,2)}$

How can I represent this on a graph?

Answer 1

I would guess the point of this is to illustrate a linear relationship between two variables.

If $f(x)$ is linear function, that means it is on the form $f(x)=ax+b$. What will this entail in terms of the quantity $f(x_i)$? It means that if you have a plot with $x_i$-values on the $x$-axis and $f(x_i)$-values on the $y$-axis, the $f(x_i)$-values will all lie on a straight line.

Now, $y_i$ is defined as $f(x_i)$ plus some error term $\epsilon_i$, which is going to spread the $y_i$-values around the straight line in a relatively regular fashion. For examples of this, just google "linear regression plot".