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Uniform distribution

Uniform Distribution

Probability Density FunctionThe general formula for the probability density function of the uniform distribution isf(x) = 1/(B - A) for A <= x <= B
where A is the location parameter and (B - A) is the scale parameter. The case where A = 0 and B = 1 is called the standard uniform distribution. The equation for the standard uniform distribution is
f(x) = 1 for 0 <= x <= 1
Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function.
The following is the plot of the uniform probability density function.
plot of the uniform probability density function
Cumulative Distribution FunctionThe formula for the cumulative distribution function of the uniform distribution isF(x) = x for 0 <= x <= 1
The following is the plot of the uniform cumulative distribution function.
plot of the uniform cumulative distribution function
Percent Point FunctionThe formula for the percent point function of the uniform distribution isG(p) = p for 0 <= p <= 1
The following is the plot of the uniform percent point function.
plot of the uniform percent point function
Hazard FunctionThe formula for the hazard function of the uniform distribution ish(x) = 1/(1-x) for 0 <= x < 1
The following is the plot of the uniform hazard function.
plot of the uniform hazard function
Cumulative Hazard FunctionThe formula for the cumulative hazard function of the uniform distribution isH(x) = -log(1 - x) for 0 <= x < 1
The following is the plot of the uniform cumulative hazard function.
plot of the uniform cumulative hazard function
Survival FunctionThe uniform survival function can be computed from the uniform cumulative distribution function.The following is the plot of the uniform survival function.
plot of the uniform survival function
Inverse Survival FunctionThe uniform inverse survival function can be computed from the uniform percent point function.The following is the plot of the uniform inverse survival function.
plot of the uniform inverse survival function
Common Statistics
Mean(A + B)/2
Median(A + B)/2
RangeB - A
Standard DeviationSQRT((B-A)**2/12)
Coefficient of Variation(B - A)/(SQRT(3)*(B + A))
Skewness0
Kurtosis9/5
Parameter EstimationThe method of moments estimators for A and B are
    A = XBAR - SQRT(3)*s
    XBAR + SQRT(3)*s
The maximum likelihood estimators are usually given in terms of the parameters aand h where
    A = a - h
    B = a + h
The maximum likelihood estimators for a and h are
    ahat = midrange(Y1, ... ,Yn)
    hhat = 0.5*[RANGE(Y1, ... , Yn)]
This gives the following maximum likelihood estimators for A and B
    Ahat = midrange(Y1, ... ,Yn) - 0.5*[RANGE(Y1, ... , Yn)]
    Bhat = midrange(Y1, ... ,Yn) + 0.5*[RANGE(Y1, ... , Yn)]
CommentsThe uniform distribution defines equal probability over a given range for a continuous distribution. For this reason, it is important as a reference distribution.One of the most important applications of the uniform distribution is in the generation of random numbers. That is, almost all random number generators generate random numbers on the (0,1) interval. For other distributions, some transformation is applied to the uniform random numbers.
SoftwareMost general purpose statistical software programs support at least some of the probability functions for the uniform distribution.

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