What does the renewal process describe?

Replacement of component.

failure time

A population of components, the failure time of each component is characterized by a non-negative random variable X. The failure time is in fact the age of the component, defining when the failure occurs.

The distribution of X could be either discrete or continuous. Discrete: X~{0,h,2h,3h,…} Continuous: the probability is determined by a probability density function (pdf) over the range of (0,$\infty$).

probability density function of X

$$f(x)=\lim_{\Delta x \to 0_+} \frac{\mathrm{prob}(x<X<x+\Delta x)}{\Delta x}$$

with

$$\int_0^{\infty} f(x) dx =1.$$

And the failure times are independent.

Other functions

cumulative distribution function $F(x)$:

$$F(x) = \mathrm{prob}(X<=x) = \int_-^x f(u) du.$$

and $f(x)=F’(x)$

survivor function $\mathscr{F}(x)$: $$\mathscr{F}(x)= \mathrm{prob(X>x)}\
= 1-F(x)\
= \int_x^{\infty} f(u) du$$

$f(x)=-\mathscr{F}'(x)$

hazard function: $\phi (x)$ the probability of almost immediate lailure of a component at age $x$.

$$\mathrm{prob} (A|B) = \frac{\mathrm{prob}(A \mathrm{and} B)}{\mathrm{prob} (B)}$$

so, $$\phi (x) = \lim_{\Delta x\to 0+} \frac{\mathrm{prob(x<X<=x+\Delta x)}}{\Delta x} \frac{1}{\mathrm{prob(x<X)}}\
= \frac{f(x)}{\mathscr{F}(x)}$$

Discrete time: Life table events

A life table consists of a list of the number of individuals, usually from an initial group of 1000 individuals so that the numbers are effectively proportions, who survive to a given age in a given population.

Important parameters:

$\mathscr{l}_x$: surviving to age $x$

$d_x$: dying between age x and x+1

$d_x = \mathscr{x}-\mathscr{x+1}$

$q_x$: those surviving to age $x$ who die before reaching age $x+1$ $q_x = d_x/\mathscr{l}_x$

neural spike train

In the presence of noise, the spike train generation is a stochastic point process, not deterministic. Hence the probability of generating the next event (spike), depends only on the “age” $t−\hat t$ of the system, i.e., the time that has passed since the last event (last spike).

The central assumption of renewal theory is that the state does not depend on earlier events .