Continuous uniform distribution

Continuous uniform distribution is a type of probability distribution that is used in statistics to describe a scenario where all outcomes are equally likely within a certain range. This distribution is characterized by two parameters: the minimum value a and the maximum value b, where a < b. The probability density function (PDF) and the cumulative distribution function (CDF) are the two main functions used to describe the behavior of the continuous uniform distribution.

Definition

The probability density function (PDF) of a continuous uniform distribution, given the parameters a and b, is defined as:

\[ f(x; a, b) = \begin{cases} \frac{1}{b-a} & \text{for } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases} \]

This function indicates that each point in the interval [a, b] has an equal chance of being drawn. The PDF is used to calculate the probability of a random variable falling within a specific range.

The cumulative distribution function (CDF), which gives the probability that a random variable X is less than or equal to a certain value x, is defined for the continuous uniform distribution as:

\[ F(x; a, b) = \begin{cases} 0 & \text{for } x < a \\ \frac{x-a}{b-a} & \text{for } a \leq x < b \\ 1 & \text{for } x \geq b \end{cases} \]

Properties

The continuous uniform distribution has several important properties:

• Mean: The mean of the distribution, which is the average of all possible values, is given by \(\mu = \frac{a + b}{2}\).
• Variance: The variance, which measures the spread of the distribution, is calculated as \(\sigma^2 = \frac{(b - a)^2}{12}\).
• Standard Deviation: The standard deviation, which is the square root of the variance, is \(\sigma = \sqrt{\frac{(b - a)^2}{12}}\).
• Skewness: The distribution is symmetric, so its skewness is 0.
• Kurtosis: The kurtosis of a continuous uniform distribution is \(-\frac{6}{5}\), indicating it is less peaked than the normal distribution.

Applications

The continuous uniform distribution is widely used in various fields such as simulation, computer science, and operations research. It is particularly useful in scenarios where a uniform random variable is needed, such as in the generation of random numbers within a specific range for simulations or modeling equally likely outcomes in games of chance.

Related Distributions

• The discrete uniform distribution is a counterpart of the continuous uniform distribution for discrete variables.
• When multiple independent variables with uniform distributions are summed, their distribution tends toward a normal distribution due to the central limit theorem.

See Also

• Probability distribution
• Normal distribution
• Binomial distribution
• Poisson distribution