science

Truncated normal distribution

Truncated normal distribution is a science 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 Truncated normal distribution rather than just read about it. In short: In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated normal distribution has wide applications in statistics and econometrics.

Key takeaways

Truncated normal 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 Truncated normal distribution to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Truncated normal distribution from memory before moving on to harder problems.

Reference excerpt

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated normal distribution has wide applications in statistics and econometrics.

Definitions Suppose X {\displaystyle X} has a normal distribution with mean μ {\displaystyle \mu } and variance σ 2 {\displaystyle \sigma ^{2}} and lies within the interval ( a , b ) , with − ∞ ≤ a < b ≤ ∞ {\displaystyle (a,b),{\text{with}}\;-\infty \leq a<b\leq \infty } . Then X {\displaystyle X} conditional on a < X < b {\displaystyle a<X<b} has a truncated normal distribution. Its probability density function, f {\displaystyle f} , for a ≤ x ≤ b {\displaystyle a\leq x\leq b} , is given by

f ( x ; μ , σ , a , b ) = 1 σ φ ( x − μ σ ) Φ ( b − μ σ ) − Φ ( a − μ σ ) {\displaystyle f(x;\mu ,\sigma ,a,b)={\frac {1}{\sigma }}\,{\frac {\varphi ({\frac {x-\mu }{\sigma }})}{\Phi ({\frac {b-\mu }{\sigma }})-\Phi ({\frac {a-\mu }{\sigma }})}}}

and by f = 0 {\displaystyle f=0} otherwise. Here,

φ ( ξ ) = 1 2 π exp ⁡ ( − 1 2 ξ 2 ) {\displaystyle \varphi (\xi )={\frac {1}{\sqrt {2\pi }}}\exp \left(-{\frac {1}{2}}\xi ^{2}\right)}

is the probability density function of the standard normal distribution and Φ ( ⋅ ) {\displaystyle \Phi (\cdot )} is its cumulative distribution function

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Truncated normal distribution

Start with the simplest possible case. Write down what Truncated normal 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 Truncated normal 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 Truncated normal 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 Truncated normal distribution

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

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How to study Truncated normal distribution in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Truncated normal 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 Truncated normal distribution out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Truncated normal distribution in simple terms?

Why does Truncated normal 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 Truncated normal 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 Truncated normal distribution.