Uniform distribution

Continuous uniform distribution

The uniform distribution on the segment [α, β] is best defined by it probability density function p(x) :

• p(x) = 0  for x < α.
• p(x) = k  for   α ≤ x ≤ β, where k is a real number.
• p(x) = 0  for  x > β.

Because the integral of a p.d.f. is 1, one must have k = 1/(β - α).

Basic properties

• The mean of the uniform distribution is

• The variance of the uniform distribution is :

• The moment generating function of the uniform distribution is :

Order statistics

We describe here the order statistics of the uniform distribution.

Estimation of the parameter of the uniform distribution U[0, θ]

Sufficient statistic

We address here the issue of identifying a sufficient statistic for the parameter θ.

Unbiased estimation

We show here that the uniform distribution does not comply with the regularity conditions needed to derive the Cramér-Rao inequality. As a consequence, we'll be able to identify an unbiased estimator of θ whose variance is smaller that the Cramér-Rao (then meaningless) "lower bound".

Discrete uniform distribution

The concept of "uniform distribution" also applies to the discrete case as well. We then have n items, and each item has the probability 1/n of being chosen during a draw. This is typically the case of a (fair !) lottery, where all the numbers have the same probability of being drawn.

The discrete uniform distribution is a special case of the multinomial distribution.

Multidimensional uniform distribution

The notion of uniform distribution extends straightforwardly to the multidimensional case. The uniform distribution in [0, 1]ⁿ is defined by its p.d.f. that is equal to :

• "1" if  0 < x_i < 1 for all   i = 1, 2, ..., n.
• "0" otherwise.

In practice, the mutidimensional uniform distribution in [0, 1]ⁿ is simulated by the product of n independent variables, all uniformly distributed in [0, 1].

The multidimensional uniform distribution in any hyper-parallelepiped whose edges are parallel to the axes is a straightforward generalisation of this distribution.

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Despite its simplicity, the uniform distribution has very important applications.

• By definition, a "random number" is the value of a random variable X whose distribution is uniform in [0, 1]. Random numbers are the cornerstone of all simulation processes (this site makes an intensive use of random numbers in its interactive animations).
  It is a remarkable fact that series of "pseudo-random numbers" can be generated by purely deterministic algorithms. In fact, for ordinary applications, it is impossible to detect the difference between a series of "truly" random numbers and a series of pseudo-random numbers.
• A consequence of the Probability Integral Transformation is that random numbers can be used for the purpose of simulating a large number of classical continuous distributions.
• The uniform distribution is also the key element of procedures for drawing :
• Random subsets from a set.
• Random permutations.
• Innocuous as it looks, the multidimensional uniform distribution is the cornerstone of one of the most important numerical computation techniques : the Monte-Carlo simulation.

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Animation : the "Broken Stick" problems

Watching the build-up of the histogram of a uniform distribution is no big thrill. So, instead, we spice up the animation by illustrating four little problems involving the uniform distribution.

Here is the first one :

• X is a uniform r.v. in [0, 1]. Its value x is represented by the black point in the illustration below.
• Once x, the value of X, is known, a r.v. Y is drawn with a uniform distribution in [x, 1]. The value of Y is represented by the red point. So the red point is drawn from the uniform distribution on the green segment.

What is the distribution of Y ? We call it the "recursive uniform" distribution (but it is sometimes less pedantically called.the "Broken Stick" distribution, for obvious reasons).

We give the answer in the Tutorial below.

We calculate :
   * Here the expectation of Y  by the Theorem of Iterated Expectation.
   * Here the variance of Y  by the Theorem of Conditional Variance.

The Broken Stick distribution is tightly connected to the distributions of the record values of the uniform distribution : we show here that it is in fact identical to that of the first upper record of the uniform distribution.

We could envision iterating the Broken Stick approach for the purpose of identifying the distribution of the second record. Unfortunately, the calculation becomes intractable. But fortunately, this distribution, as well as that of the records of any order, can be found by a more general method that we then describe.

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In the same animation, we also illustrate three other distributions labeled "Short", "Long" and "Random".

As before, X is uniform in [0, 1]. But now, instead of choosing Y uniformly in the right-hand side segment [x, 1] :

• If "Short" is selected, Y is uniform in the shorter of the two segments [0, x] and [x, 1]. What is the distribution of Y ?
• If "Long" is selected, Y is uniform in the longer of the two segments [0, x] and [x, 1]. What is the distribution of Y ?
• If "Random" is selected, either [0, x] or [x, 1] is selected randomly with probability 0.5 for each. Y is then uniformly drawn in the selected segment. What is the distribution of Y ?

We give the answer in the Tutorial below.

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If you solved these problems, you'll find it easy to solve these last two. They are variants of the "Random" case. After X has been drawn, the segment in which Y is going to be drawn is still selected at random. But instead of assigning a 0.5 probability to each of the segments :

• In the first problem :

        * The left-hand side segment is selected with probability x,   while

        * The right-hand side segment is selected with probability 1 - x.

In other words, each segment is assigned a probability to be selected equal to its length.
 

What is the distribution of Y ?

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• In the second problem :

        * The left-hand side segment is selected with probability 1 - x,   while

        * The right-hand side segment is selected with probability x.
 

What is the distribution of Y ?

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We encourage you to solve these two problems, as the solutions will probably surprise you.

Product and ratio of uniform random variables

We calculate here the distributions of :

    * The product,

    * And the ratio

of two independent and uniformly distributed random variables as an illustration of the concept of "marginal distribution".

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In this Tutorial, we establish the elementary properties of the uniform distribution. Because of the simplicity of the p.d.f., the most convenient way to calculate the moments is straight calculation from the definition of moments.

Yet, it is quite instructive to also calculate these moments with the moment generating function. Although the function itself is simple enough, it and all its derivatives are in the 0/0 form for t = 0, thus making a straightforward identification of the moments impossible. We go around this difficulty using two methods : Taylor expansion and L'Hôpital rule.

So, although the m.g.f. is not the most appropriate tool for calculating the moments of the uniform distribution, it gives us an opportunity to go over some useful mathematical techniques.

BASIC PROPERTIES OF THE UNIFORM DISTRIBUTION

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In this rather lengthy Tutorial, we go over some important applications of the uniform distribution.

• Generating random integers from the continuous uniform distribution in [0, 1].
• We then spend some time on a somewhat unsual problem : an urn contains a very large number of balls of various colors, and its is asked to estimate the number of colors represented in the urn without assessing the number of balls for each and every color. The solution relies heavily on the discrete uniform distribution.
• We finally address the question of drawing :
• random subsets from a set,  
• random permutations.of ordered set.

We review some of the many solutions to these classical and important problems. Some demonstrations are left as exercises.

SOME APPLICATIONS OF THE UNIFORM DISTRIBUTION

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We give here the solutions to the first four problems.

SOLUTIONS TO THE FIRST FOUR PROBLEMS

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Related readings :