Skip to main content

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.

Comments

Post a Comment

Popular posts from this blog

Double exponential distribution

Double Exponential Distribution Probability Density Function The general formula for the  probability density function  of the double exponential distribution is where   is the  location parameter  and   is the  scale parameter . The case where   = 0 and   = 1 is called the  standard double exponential distribution . The equation for the standard double exponential distribution is 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 double exponential probability density function. Cumulative Distribution Function The formula for the  cumulative distribution function  of the double exponential distribution is The following is the plot of the double exponential cumulative distribution function. Percent Point Function The formula for the  percent point function  of the double exponential distribution
Post a Comment
Read more

Runs Test for Detecting Non-randomness

Runs Test for Detecting Non-randomness Purpose: Detect Non-Randomness The runs test ( Bradley, 1968 ) can be used to decide if a data set is from a random process. A run is defined as a series of increasing values or a series of decreasing values. The number of increasing, or decreasing, values is the length of the run. In a random data set, the probability that the ( I +1)th value is larger or smaller than the I th value follows a binomial distribution , which forms the basis of the runs test. Typical Analysis and Test Statistics The first step in the runs test is to count the number of runs in the data sequence. There are several ways to define runs in the literature, however, in all cases the formulation must produce a dichotomous sequence of values. For example, a series of 20 coin tosses might produce the f
Post a Comment
Read more

Basics of Sampling Techniques

Population                A   population   is a group of individuals(or)aggregate of objects under study.It is also known as universe. The population is divided by (i)finite population  (ii)infinite population, (iii) hypothetical population,  subject to a statistical study . A population includes each element from the set of observations that can be made. (i) Finite population : A population is called finite if it is possible to count its individuals. It may also be called a countable population. The number of vehicles crossing a bridge every day, (ii) Infinite population : Sometimes it is not possible to count the units contained in the population. Such a population is called infinite or uncountable. ex, The number of germs in the body of a patient of malaria is perhaps something which is uncountable   (iii) Hypothetical population : Statistical population which has no real existence but is imagined to be generated by repetitions of events of a certain typ
Post a Comment
Read more