likelihood function of a line regression equation

Question

I am a bit confused at how can i find the likelihood function and the solutions of likelihood function for a line equation, for example y=3x+15

Answer 1

The equation $y = 0.2x + 4$ is not a statistical model, but a mathematical equation. It doesn't even have parameters (unless you mean that $x$ is a parameter), so there is nothing to optimize. In case you meant $y = \beta_0 + \beta_1 x$, this would still be a deterministic function of the variables, that can be solved by pure algebra, that does not need statistics. Statistical model arises, when there is uncertanity about the data

$$ y = \beta_0 + \beta_1 x \color{red}{+ \varepsilon} $$

Assuming that by linear regression you mean the standard model minimizing the squared error (e.g. with ordinary least squares), then the objective function to minimize is

$$ \operatorname{SSE}(y, \hat y) = \sum_{i=1}^n (y_i - \hat y_i)^2 $$

To obtain a likelihood function, you would need a probability distribution. Usually we maximize the log-likelihood (or minimize the negative log-likelihood), rather then pure likelihood. The distribution that is equivalent to minimizing squared errors would be

$$ \log p(y) \propto \sum_{i=1}^n -\frac{(y_i - \hat y_i)^2}{\sigma^2} $$

where the proportionality ($\propto$) is due the fact, that for optimization we can drop the normalizing constant (the distribution does not have to integrate to one). As you can see, we introduced a new parameter for $\sigma^2$ for the variance of the residuals. If you take $\exp$ of it, to reverse the $\log$ we took, and get rid of the negative sign, this is a normal distribution

$$ p(y) \propto \exp\Big(-\frac{(y-\mu)^2}{2\sigma^2}\Big) $$

So, minimizing sum of squared errors is equivalent to maximizing the Gaussian likelihood. The above is a different way of saying, that the linear regression model assumes normally distributed noise $\varepsilon$ that has mean of zero,

$$\begin{align} y &= \beta_0 + \beta_1 x + \varepsilon \\ \varepsilon &\sim \mathcal{N}(0, \sigma^2) \end{align}$$