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Bernoulli distribution is the discrete likelihood distribution of a variable that takes a Boolean or binary output. The notion is if you are performing an experiment that can either end in success or failure, you can associate a probability p with success labeled with 1 and have a probability (1-p) with failure labeled with 1. The parameter of the Bernoulli distribution is the probability of success p. The Bernoulli distribution experiment is only repeated once.
From your standard calculus and differential equation class, your professor must have mentioned that the main area of study is the derivative of a function. Stochastic integration is based on Brownian motion and requires limiting the class of possible integrand to processes that have been adapted. The Ito convention states that a stochastic integral that takes into account the concept of Brownian motions should be a martingale. Stochastic integrals are defined the same way as the Riemann integral.
A Poisson process models a series of discrete events. In this type of model, while the estimated period between occurring events is known, the precise timing is random. An event doesn’t interfere with the arrival of the next event. Meaning, the waiting time of events is memoryless. Poisson processes should meet the following assumptions:
- All events should be independent of each other
- Two occurrences cannot happen simultaneously
- The phenomena per period are constant
Named after Andrey Markov, Markov processes are random processes, in which the probability of an outcome of an event is based on the results of previous events. Markov processes are applied in many different areas in real life including studying and evaluating cruise control systems in vehicles, queuing customers who are arriving at an airport, analyzing currency exchange rates, and studying animal population dynamics to mention a few.