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Properties of Uniform Distribution

Definition

The most basic form of continuous probability distribution function is called the uniform distribution. It is a rectangular distribution with constant probability and implies the fact that each range of values that has the same length on the distributions support has equal probability of occurrence.  This belongs to the category of maximum entropy probability distributions.

The probability density function for the variable x given that a ≤ x ≤ b is given by:

ud1

Characteristics

The following are the key characteristics of the uniform distribution:

  • The density function integrates to unity
  • Each of the inputs that go in to form the function have equal weighting
  • Mean of the unifrom function is given by:

ud2

  • The variance is given by the equation:

ud3

The plot of the uniform function is as below:

ud4

The location of the interval has little influence in deciding if the uniformly distributed variable falls within the fixed length. Two factors that influence this the most are the interval size and the fact that the interval falls within the distributions support.

  • Standard uniform distribution: If a =0  and b=1 then the resulting function is called a standard unifrom distribution. This has very important practical applications.
  • There are variables in physical, management and biological sciences that have the properties of a uniform distribution and hence it finds application is these fields.

Example

Let’s say the amount of gasoline sold every day at a service station is uniformly distributed. The minimum sold is 1000 gallons and maximum sold is 3000 gallons.

We can calculate the probability that the service station will sell atleast 2,000 gallons using the uniform distribution properties.

ud5

P(X ≥ 2,000) = (3,000 – 2,000) x  (1/2000)   = .50

Applications

  • The most common use of the uniform distribution is as a starting point for the process of random number generation. A classic example of this would be in programming languages.
  • It is most commonly used for sampling arbitrary distributions. The most general method used is the inverse transform sampling method which uses the cumulative distribution function of the target random variable. There is very wide application in the theoretical space in this case. Inverting the CDF is a very important requirement in this process and when the CDF is not known in closed form alternative methods have been invented of which one is called rejection sampling. A prime example of this is the use of the normal distribution method to get over the inefficiences of the inverse transform technique.
  • When analog signals are converted to digital format errors happen due to rounding or truncation which are called quantization errors. In cases where the error does not have significant correlation with the signal this is uniformly distributed.
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