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Continuous Distributions

Main Concept

Random Variables and Probability Distributions

 

A random variable is a property of a random process. That is, a process in which it is difficult to predict the outcome. The process may occur many times, and each time, the random variable may have a different value.

 

A random variable is called discrete if it can have only countably many possible values. In other words, the set of possible values can be listed, even if this listing continues forever.

 

Otherwise, the random variable is continuous. Typically, the set of possible values for a continuous random variable forms a range.

 

For example, the rolling of a die can be considered a random process. The number that shows on the top of the die after the roll is a discrete random variable associated to that process, and the amount of time it takes for the die to stop rolling is an associated continuous random variable.

 

Each random variable X has an associated a probability distribution, which describes, for each value that X can have, the probability that X will actually have that value (or in the case of continuous distributions, a value close to it).

 

Continuous distributions can be represented by two types of functions:

 

Probability density function (PDF): The value of the PDF at any point x represents the limiting value of probability that the value of X lies within a range containing x, divided by the size of the range, as the size of the range approaches 0.

 

Cumulative distribution function (CDF): The value of the CDF at a point x represents the probability that the value of X is less than or equal to x.

 

Note that the PDF is the derivative of the CDF. The CDF asymptotically approaches 1 as x goes to infinity, while the integral of the PDF over all possible values equals 1.

 

Below is a list of the most common continuous distributions:

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Beta distribution

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Cauchy distribution

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Chi-square distribution

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Erlang distribution

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Error (exponential power) distribution

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Exponential distribution

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Fisher f-distribution

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Gamma distribution

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Gumbel distribution

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Inverse Gaussian (Wald) distribution

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Laplace distribution

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Logistic distribution

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Log normal distribution

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Maxwell distribution

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Moyal distribution

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Normal (Gaussian) distribution

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Pareto distribution

• 

Power distribution

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Rayleigh distribution

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Student-t distribution

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Triangular distribution

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Uniform (rectangular) distribution

• 

Von Mises distribution

• 

Weibull distribution

 

Change the distribution and adjust the parameters to see the graphs of the associated probability density and cumulative distribution functions.

 

Distribution

Range 

Parameters

 

 

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