Define the Ordinary, modified and equilibrium renewal processes with examples. Also, differentiate between modified and equilibrium renewal processes.

In the context of stochastic processes, renewal processes are used to model situations where events occur randomly over time, but the times between successive events (called inter-arrival times) are independent and identically distributed (i.i.d.). There are three primary types of renewal processes: the ordinary renewal process, the modified renewal process, and the equilibrium renewal process. Each has unique characteristics and applications.

### Ordinary Renewal Process

The ordinary renewal process is the simplest form of a renewal process. In this process, events occur at random time points, with the times between successive events (denoted as inter-arrival times) being independent and identically distributed random variables. The key characteristic of the ordinary renewal process is that it starts from an initial state, typically at time zero, and the first event occurs after a random amount of time, with subsequent events occurring at random intervals.

The time between successive events is governed by a probability distribution, often referred to as the "renewal distribution." For example, if the inter-arrival times follow an exponential distribution, the process is known as a Poisson process, which is a specific type of ordinary renewal process.

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