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Ratio of uniforms

Ratio of uniforms is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Ratio of uniforms rather than just read about it. In short: The ratio of uniforms is a method initially proposed by Kinderman and Monahan in 1977 for pseudo-random number sampling, that is, for drawing random samples from a statistical distribution. Like rejection sampling and inverse transform sampling, it is an exact simulation method.

Ratio of uniforms — main illustration
Ratio of uniforms — illustration

Key takeaways

  • Ratio of uniforms belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Ratio of uniforms to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Ratio of uniforms from memory before moving on to harder problems.

Reference excerpt

The ratio of uniforms is a method initially proposed by Kinderman and Monahan in 1977 for pseudo-random number sampling, that is, for drawing random samples from a statistical distribution. Like rejection sampling and inverse transform sampling, it is an exact simulation method. The basic idea of the method is to use a change of variables to create a bounded set, which can then be sampled uniformly to generate random variables following the original distribution. One feature of this method is that the distribution to sample is only required to be known up to an unknown multiplicative factor, a common situation in computational statistics and statistical physics.

Motivation

A convenient technique to sample a statistical distribution is rejection sampling. When the probability density function of the distribution is bounded and has finite support, one can define a bounding box around it (a uniform proposal distribution), draw uniform samples in the box and return only the x coordinates of the points that fall below the function (see graph). As a direct consequence of the fundamental theorem of simulation, the returned samples are distributed according to the original distribution. When the support of the distribution is infinite, it is impossible to draw a rectangular bounding box containing the graph of the function. One can still use rejection sampling, but with a non-uniform proposal distribution. It can be delicate to choose an appropriate proposal distribution, and one also has to know how to efficiently sample this proposal distribution. The method of the ratio of uniforms offers a solution to this problem, by essentially using as proposal distribution the distribution created by the ratio of two uniform random variables.

Statement The statement and the proof are adapted from the presentation by Gobet

Complements

Rejection sampling in A f , r {\displaystyle A_{f,r}}

… excerpt ends here. Continue reading the full article.

Illustrations

Ratio of uniforms: Exponential distribution before and after change of variables by the ratio of uniforms method. Top: graph of the exponential distribution on R + {\displaystyle \mathbb {R} ^{+}} . Bottom: the set A f , 1 {\displaystyle A_{f,1}} is represented in the space ( u , v ) {\displaystyle (u,v)} , inscribed in the bounding box A ~ f , 1 {\displaystyle {\tilde {A}}_{f,1}} . The colored domains, of equal probability, were added to help the visual association of the corresponding domains of the transformed sets.
Exponential distribution before and after change of variables by the ratio of uniforms method. Top: graph of the exponential distribution on R + {\displaystyle \mathbb {R} ^{+}} . Bottom: the set A f , 1 {\displaystyle A_{f,1}} is represented in the space ( u , v ) {\displaystyle (u,v)} , inscribed in the bounding box A ~ f , 1 {\displaystyle {\tilde {A}}_{f,1}} . The colored domains, of equal probability, were added to help the visual association of the corresponding domains of the transformed sets.
Ratio of uniforms: Normal mixture distribution before and after change of variables by the ratio of uniforms method. Top: graph of the mixture distribution on R {\displaystyle \mathbb {R} } . Bottom: the set A f , r {\displaystyle A_{f,r}} is represented for two different values of r {\displaystyle r} . The solid lines on the top represent the de-transformation of the bounding boxes on the bottom. The solid lines on the bottom represent the locations of different values of x {\displaystyle x} in the set.
Normal mixture distribution before and after change of variables by the ratio of uniforms method. Top: graph of the mixture distribution on R {\displaystyle \mathbb {R} } . Bottom: the set A f , r {\displaystyle A_{f,r}} is represented for two different values of r {\displaystyle r} . The solid lines on the top represent the de-transformation of the bounding boxes on the bottom. The solid lines on the bottom represent the locations of different values of x {\displaystyle x} in the set.

Worked examples

Example 1 — a first encounter with Ratio of uniforms

Start with the simplest possible case. Write down what Ratio of uniforms claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Ratio of uniforms before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Ratio of uniforms ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Ratio of uniforms

In research
Ratio of uniforms appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Ratio of uniforms in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Ratio of uniforms is common in secondary-school and first-year university syllabi. It links to neighbouring topics Non-uniform random numbers, Pseudorandom number generators, so understanding it makes those chapters shorter.
In everyday life
Look for Ratio of uniforms outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Ratio of uniforms in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Ratio of uniforms means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Ratio of uniforms out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Ratio of uniforms in simple terms?

The ratio of uniforms is a method initially proposed by Kinderman and Monahan in 1977 for pseudo-random number sampling, that is, for drawing random samples from a statistical distribution. Like rejection sampling and inverse transform sampling, it is an exact simulation method.

Why does Ratio of uniforms matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Ratio of uniforms?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Ratio of uniforms.

Tags

  • Non-uniform random numbers
  • Pseudorandom number generators

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