Probability distribution

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A probability distribution describes a special universe, a set of real numbers (see analysis) and how probability is distributed among them to determine a random variable. For every random variable, there is a function called the cumulative distribution function which provides the probability that a given value is not exceeded by the random variable.

    F(x) = Pr[X<=x] for x in the domain of the random variable

If the random variable is a discrete random variable, all of the probability is concentrated on a discrete set of points. We can define the probability for a specific point by the probability mass function.

    p(x) = limit{F(x+t)-F(x-t)} as t goes to zero.

For a continuous random variable, we define the probability density function (at x) by

           dF(x)         F(x+t) - F(x)
    f(x) = ----- = limit ------------- as t goes to zero.
            dx                   t

Several probability distributions are so important that they have been given specific names, the normal distribution, the binomial distribution, the Poisson distribution are just three of them.

probability axioms -- probability applications
random variable -- cumulative distribution function -- probability density

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