Geometric distribution, discrete probability distributions

Geometric Probability distribution of the number of failures before tlie first success occurs. It is the discrete analog of the exponential distribution, where parameter p is analogous to Xc. Distribution assumes inemoryless property of independent trials Can be applied to discrete failure on demand data in absence of other information... 

The next step is to evaluate the adsorption residence time resulting from a series of encounters. Evidently, the number of short displacements, and also the adsorption events in a sequence, has the discrete geometric probability distribution... 

Exponential Distribution A third probability distribution arising often in computational biology is the exponential distribution. This distribution can be used to model lifetimes, analogous to the use of the geometric distribution in the discrete case as such it is an example of a continuous waiting time distribution. A r.v. X with this distribution [denoted by X exp(A)] has range [0, - - >] and density function... 

Now let and k be arbitrarily large. Then, for a fixed volume, m, the probability (8) is invariant except for the upper limit of the product. This is exactly equivalent to the geometric distribution whose discrete probability density is given by... 

Both the RBC distribution (8) and the geometric distribution (11) are defined only for specific integer bubble sizes, and derivatives of their distribution functions do not exist. For subsequent developments we need an equivalent continuous distribution. Fortunately, for N and k large with respect to m, both discrete distributions can be closely approximated by the exponential distribution if its mean is set to the RBC mean volume given by (10). The exponential probability density is... 

Geometric Distribution n A probability distribution of a discrete random variable, X, with non-negative integer values, k, that have a probability, P(/c), of the form ... 

It is helpful to have standard probabihty models that are useful for analyzing large biological data, in particular bioinformatics. There are six standard distributions for discrete r.v. s, that is, BemouUi for binary r.v. s, (e.g., success or failure), binomial for the number of successes in n independent BemouUi trials with a common success probabihty p, uniform for model simations where aU integer outcomes have the same probabihty over an interval [a, b, geometric for the number of trials required to obtain the first success in a sequence of independent BemouUi trials with a common success probabihty p, Poisson used to model the number of occurrences of rare events, and negative binomial for the number of successes in a fixed number of Bemoulh trials, each with a probability p of success. 

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