# Likelihood function for a unit $i$ with death at time $t_{i}$ and non - death at time $t_{i}$

The survival function with respect to time is defined to be

$$S(t) = 1 - F(T) = \int_{t}^{\infty}f(x)dx$$

With a bit of algebraic manipulation, one arrives at the time dependent hazard rate function,

$$\lambda(t) = \frac{f(t)}{S(t)}$$

which says that

the probability density function $$f(t)$$ of the cumulative probability density function $$F(t)$$ is

$$f(t) = \lambda(t) S(t)$$

Now, suppose there exists $$n$$ samples to be observed for a duration $$t_{i}$$ where $$i$$ is the i-th sample.

If a death occurs at time $$t_{i}$$, then the likelihood function is described to be

$$L_{i} = f(t_{i}) = S(t_{i})\lambda(t_{i})$$

However, if a death does not occur, the likelihood function is described to be

$$L_{i} = f(t_{i}) = S(t_{i})$$ where $$\lambda(t) = 1$$.

What is the significant of the physical intuition behind the fact that $$\lambda(t) = 1$$? Mathematically, $$\lambda(t) = 1$$ implies that $$f(t) = S(t)$$ to suggest that the probability density function for an occurrence of death for sample $$i$$ intersects with the survivability function S(t) for sample $$i$$.

Any help to further illuminate my doubts is greatly appreciated.