so far the likelihood function was defined as follows: $$ L(\theta) = \prod_i f(x_i), $$ where $f$ is the density of random variable $X$.
Ok, but there comes survival analysis. And at the lecture it is rewritten (defined?) differently $$ L(\theta) = \prod_i f(t_j;\theta)^{\delta_j} S(t_j;\theta)^{1-\delta_j}. $$ Here $t_j$ is the time of event, $\delta_j$ -- indicator it the event was censored or not. $S(t) = P(T>t)$, survival function for the random variable $T$ of the event of the time. Why? Where does this equation come from? Why when i deal with densities suprisingly the cdf appears?
The likelihood is the joint probability distribution of the random sample. Namely, if you observed $x$, that is, $X = x$, namely the random variable $X$ has been realized at some point $x$ that you observed. Assume that you observe the lifetime of something and you know that this something died at time $x$, thus $x$ is the realization of $X$. Now assume that you also observe the lifetime of something else that is distributed according to some r.v. $Y$. However, at some time point $t$ the observation was censored before it had been realized, that is, instead of observing $Y=y$, you only know that at time $t$ this thing was still alive. Assuming that $X$ and $Y$ are independent, what is their joint density? $$ f_{X,I\{Y>t\}}(x,i)=f_X(x)\mathbb{P}(Y>t)=f_X(x)S(t), $$
now, you can generalize it for $n$ observations that for each one either it was censored, hence you only know that $X_i>t$ or realized, hence you know that $X_i=x_i$. Where $\delta_i$ just indicates which of the scenario you observed for the particular r.v. $X_i$.