Uniform Distribution

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Discrete uniform distribution

A discrete random variable X with a finite number of outcomes is called uniform distribution, if each value of X can occur with an equal probability, which depends on n. The probability density function of a uniform random variable is:

The distribution function for a uniform random variable is:

The expected value and variance of discrete uniform random variable X are:

Continuous uniform distribution

A continuous random variable X on the interval [a,b] has a uniform distribution if each point in that interval has an equal probability of occurring, the density function will have the following form:

The distribution function for a continuous uniform random variable is: The expected value and variance of continuous uniform random variables are: The parameters of a continuous uniform distribution are and .

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A man arrives at a tram stop, but does not know the schedule of the tram. The tram arrives at that stop every 20 minutes. Define the random variable X: ”waiting time for a tram in minutes”. This random variable can take any value in the interval [0,20]. This implies: P(0 X 20) = 1, a = 0, b = 20. The random variable waiting time will have a uniform distribution. Density of X: Distribution function: The expected value of X is:

On average the a person will have to wait 10 minutes for a tram. The variance is:

Standard deviation: = 5.77 .The density and the distribution function look like this:

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Discrete Uniform Distribution

The probability density function of discrete Uniform random variable can be illustrated with a histogram. The distribution function of this random variable, on the other hand, will be a step function. A common example of a discrete Uniform random variable are the outcomes associated with the roll of a fair die. The discrete random variable X (= result of the throw) can take integer numbers between 1 and 6. If the dice are “fair”, the probability of each outcome of X is

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Continuous Uniform Distribution

Let us verify whether is a density function: First, , so f(x) 0 for all x, i.e. the function is nonnegative. Furthermore we have: This indicates that f(x) is a density. The distribution function F(x) can be computed as: The expected value and the variance for this random variable are: The following diagram illustrates the density and distribution function of a continuous Uniform random variable. Density:

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Distribution function:

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