Probability distribution

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.

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For Continous Random Variable you ARE giving the Cumulative Distribution Function, not the Probability Density Function...eh?

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The cdf is defined for all random variables, discrete or continuous, so it is a better starting point than either the probability function or the density function. In one case you use differences to get the probability function and in the other you use the derivative. Most students are introduced to the derivative before the integral, so this approach is a bit more accessible -- DickBeldin

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The probability that a continuous random variable X takes a value less than or equal to x is denoted Pr(X<=x). The probability density function of X, where X is a continuous random variable, is the function f such that

• Pr(a<=X<=b)= INTEGRAL ( as x ranges from a thru b) f(x) dx.

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Correct, but F[b]-F[a] gives the probability of an interval directly without all the complications. We hide the complications in the cdf. It is inconvenient that we can't feature the explicit form of the cdf for many of the distributions we like to use, but it is important to build the concepts with proper spacing of the difficulties. One hurdle, then a straight stretch, then a curve, then another straight ... --DickBeldin

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You may present this material as you feel best. I don't disagree with your argument. But, mislabelling definitions is never okay. You have defined the probability density function for continuous random variables with the cumulative distribution function for the same. RoseParks

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Probability Axioms -- Probability Applications

Random Variable -- Cumulative Distribution Function -- [Probability Density]?

back to Probability and Statistics

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