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Stochastic Process

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Given a probability space and a measurable space , a stochastic process is a family of stochastic variables , that is a map

,

such that for all the map is --measurable.

If is finite or countable, is called a point process.


Example: Poisson Process

Point Process

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A poisson process is a counting process, that is a stochastic process {N(t), t ≥ 0} with values that are positive, integer, and increasing:

  1. N(t) ≥ 0.
  2. N(t) is an integer.
  3. If st then N(s) ≤ N(t).

Poisson Distribution

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The poisson distribution of intensity of a stochastic variable , is a probability distribution given by the probability mass function

For the poisson distribution to be a well-defined distribution, we need to check that . Indeed,

Then, also, exists for every subset , since is bounded by one and a monotonic growing function in , since is positive for all .

The expected value of a stochastic variable X following poisson distribution is computed as (link Expactation value of a discrete random variable) :

Expactation value of a discrete random variable

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Let be a discrete stochastic variable. Then the expected value of can be calculated as

Proof:

It is for , we have

.

Thus

Binomial Distribution

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The binomial distribution with parameters n and p of a stochastic variable , is a probability distribution of X given by the probability mass function

If X follows the binomial distribution with parameters n, the number of independent experiments, and p, the probability for one experiment to give the answer "yes", we write K ~ B(np).

We have

The expected value of a stochastic variable X following the binomial distribution is calculated as (link Expactation value of a discrete random variable) :

Its variance is given by

where we used the computational formula for the variance in . (uncomplete proof!)

Spiketrains and instanteous firing rate (article)

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Reference: Poisson Model of Spike Generation - David Heeger


A spike train of n spikes occuring at times , is given as function

which is more sophistically known as the neural response function.

The number of spikes N, occuring between two points in time , is computed as

Because the sequence of action potentials generated by a given stimulus typically varies from trial to trial, neuronal responses are typically treated probabilistically. One (very simple) way to characterize the probabilitistic behaviour of the firing of a neuron is by the spike count rate r, which is given by

The spike count rate be determined vor a single trial period, or can be averaged over several trials. Another possible way of characterization