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Uniform distribution
The uniform distribution on the segment [α, β] is best defined by it probability density function p(x) :
Because the integral of a p.d.f. is 1, one must have k = 1/(β - α).
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µ = (α + β)/2 |
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Var = (β - α)²/12 |
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We describe here the order statistics of the uniform distribution.
* U[0, θ]
We show here that X(n), the largest observation in a n-sample, is a sufficient statistic for θ. We'll then show that this statistic is in fact minimal sufficient, then that it is complete.
* U[θ, θ + 1]
We'll identify a sufficient statistic for θ, then show that this statistic is minimal sufficient. This is an example of a minimal sufficient statistic whose dimension does not match that of the parameter to be estimated.
We'll then show that the statistic is not complete from which we'll deduce that in such a setting, θ has no complete statistic.
* U[0, θ]
We show here that the uniform distribution U[0, θ] 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 than the (then meaningless) Cramér-Rao "lower bound".
In fact, we show here that this estimator is the UMVUE of θ, a result that we'll then generalize by identifying the UMVUE of any differentiable function of θ.
Note, though, that we identify here
a biased estimator of θ whose Mean Squared Error (MSE) is
less than that of the UMVUE of θ.
* U[θ, θ + 1]
We show here that no function of θ admits a UMVUE.
Tests on θ
We build here the Likelihood Ratio Test testing the null hypothesis θ = θ0 against the alternative hypothesis θ ≠ θ0 for the uniform distribution U[0, θ].
The same scheme may be used for testing other null and alternative hypothesis.
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.
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 :
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.
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 :
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.
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The "Book of Animations" on your computer |
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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] :
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 :
* 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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* 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.
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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Tutorial 1 |
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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Cumulative distribution function Probability for the value of X to be in any interval All order moments Mean Variance Higher order moments Moment generating function and moments Moment generating function Moments Taylor expansion L'Hôpital rule |
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TUTORIAL |
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Tutorial 2 |
In this rather lengthy Tutorial, we go over some important applications of the uniform distribution.
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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The discrete uniform distribution The "How many colors ?" problem The problem Ball multiplicities and number of colors Multiplicities m(i) Number of colors as a function of multiplicities Estimating the number of colors Random subsets The notion of "random subset" Two intuitive methods Enumerating k-subsets Drawing without replacement (No demonstration) A recursive method based on conditional probabilities General principle Probability for the first element to be in the k-subset The general case The recursive algorithm Tree-like representation Random permutations Enumerating permutations Exhaustive draw without replacement (no demonstration) |
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TUTORIAL |
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Tutorial 3 |
We give here the solutions to the first four problems.
SOLUTIONS TO THE FIRST FOUR PROBLEMS
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The "Broken stick" distribution The "Short" problem Preliminary remark Left-hand side distribution Right-hand side distribution The "Long" problem Left-hand side distribution Right-hand side distribution The "Random" problem Mixture of random variables The solution |
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TUTORIAL |
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Related readings :
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Download this Glossary |
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