Trapezoidal distribution

Key takeaways

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

Reference excerpt

In probability theory and statistics, the trapezoidal distribution is a continuous probability distribution whose probability density function graph resembles a trapezoid. Likewise, trapezoidal distributions also roughly resemble mesas or plateaus. Each trapezoidal distribution has a lower bound a and an upper bound d, where a < d, beyond which no values or events on the distribution can occur (i.e. beyond which the probability is always zero). In addition, there are two sharp bending points (non-differentiable discontinuities) within the probability distribution, which we will call b and c, which occur between a and d, such that a ≤ b ≤ c ≤ d. The image to the right shows a perfectly linear trapezoidal distribution. However, not all trapezoidal distributions are so precisely shaped. In the standard case, where the middle part of the trapezoid is completely flat, and the side ramps are perfectly linear, all of the values between c and d will occur with equal frequency, and therefore all such points will be modes (local frequency maxima) of the distribution. On the other hand, though, if the middle part of the trapezoid is not completely flat, or if one or both of the side ramps are not perfectly linear, then the trapezoidal distribution in question is a generalized trapezoidal distribution, and more complicated and context-dependent rules may apply. The side ramps of a trapezoidal distribution are not required to be symmetric in the general case, just as the sides of trapezoids in geometry are not required to be symmetric. The non-central moments of the trapezoidal distribution are

E ⁡ [ X k ] = 2 d + c − b − a 1 ( k + 1 ) ( k + 2 ) ( d k + 2 − c k + 2 d − c − b k + 2 − a k + 2 b − a ) {\displaystyle \operatorname {E} [X^{k}]={\frac {2}{d+c-b-a}}{\frac {1}{(k+1)(k+2)}}\left({\frac {d^{k+2}-c^{k+2}}{d-c}}-{\frac {b^{k+2}-a^{k+2}}{b-a}}\right)}

Special cases of the trapezoidal distribution include the uniform distribution (with a = b and c = d) and the triangular distribution (with b = c). Trapezoidal probability distributions seem to not be discussed very often in the literature. The uniform, triangular, Irwin-Hall, Bates, Poisson, normal, bimodal, and multimodal distributions are all more frequently discussed in the literature. This may be because these other (non-trapezoidal) distributions seem to occur more frequently in nature than the trapezoidal distribution does. The normal distribution in particular is especially common in nature, just as one would expect from the central limit theorem.

See also Trapezoid Probability distribution Central limit theorem Uniform distribution (continuous) Triangular distribution Irwin–Hall distribution Bates distribution Normal distribution Multimodal distribution Poisson distribution

References

Illustrations

Worked examples

Example 1 — a first encounter with Trapezoidal distribution

Start with the simplest possible case. Write down what Trapezoidal distribution claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Trapezoidal distribution 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 Trapezoidal distribution 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 Trapezoidal distribution

In research: Trapezoidal distribution appears in science 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 Trapezoidal distribution 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: Trapezoidal distribution is common in secondary-school and first-year university syllabi. It links to neighbouring topics Continuous distributions, so understanding it makes those chapters shorter.
In everyday life: Look for Trapezoidal distribution 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.

How to study Trapezoidal distribution in 20 minutes

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

Frequently asked questions

What is Trapezoidal distribution in simple terms?

In probability theory and statistics, the trapezoidal distribution is a continuous probability distribution whose probability density function graph resembles a trapezoid. Likewise, trapezoidal distributions also roughly resemble mesas or plateaus.

Why does Trapezoidal distribution matter?

Because it connects several science 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 Trapezoidal distribution?

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 Trapezoidal distribution.