Negative multinomial distribution

Key takeaways

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

Reference excerpt

In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(x0, p)) to more than two outcomes. As with the univariate negative binomial distribution, if the parameter x 0 {\displaystyle x_{0}} is a positive integer, the negative multinomial distribution has an urn model interpretation. Suppose we have an experiment that generates m+1≥2 possible outcomes, {X0,...,Xm}, each occurring with non-negative probabilities {p0,...,pm} respectively. If sampling proceeded until n observations were made, then {X0,...,Xm} would have been multinomially distributed. However, if the experiment is stopped once X0 reaches the predetermined value x0 (assuming x0 is a positive integer), then the distribution of the m-tuple {X1,...,Xm} is negative multinomial. These variables are not multinomially distributed because their sum X1+...+Xm is not fixed, being a draw from a negative binomial distribution.

Properties

Marginal distributions If m-dimensional x is partitioned as follows

X = [ X ( 1 ) X ( 2 ) ] with sizes [ n × 1 ( m − n ) × 1 ] {\displaystyle \mathbf {X} ={\begin{bmatrix}\mathbf {X} ^{(1)}\\\mathbf {X} ^{(2)}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}n\times 1\\(m-n)\times 1\end{bmatrix}}}

and accordingly p {\displaystyle {\boldsymbol {p}}}

p = [ p ( 1 ) p ( 2 ) ] with sizes [ n × 1 ( m − n ) × 1 ] {\displaystyle {\boldsymbol {p}}={\begin{bmatrix}{\boldsymbol {p}}^{(1)}\\{\boldsymbol {p}}^{(2)}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}n\times 1\\(m-n)\times 1\end{bmatrix}}}

and let

q = 1 − ∑ i p i ( 2 ) = p 0 + ∑ i p i ( 1 ) {\displaystyle q=1-\sum _{i}p_{i}^{(2)}=p_{0}+\sum _{i}p_{i}^{(1)}}

The marginal distribution of X ( 1 ) {\displaystyle {\boldsymbol {X}}^{(1)}} is N M ( x 0 , p 0 / q , p ( 1 ) / q ) {\displaystyle \mathrm {NM} (x_{0},p_{0}/q,{\boldsymbol {p}}^{(1)}/q)} . That is the marginal distribution is also negative multinomial with the p ( 2 ) {\displaystyle {\boldsymbol {p}}^{(2)}} removed and the remaining p's properly scaled so as to add to one. The univariate marginal m = 1 {\displaystyle m=1} is said to have a negative binomial distribution.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Negative multinomial distribution

Start with the simplest possible case. Write down what Negative multinomial 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 Negative multinomial 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 Negative multinomial 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 Negative multinomial distribution

In research: Negative multinomial 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 Negative multinomial 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: Negative multinomial distribution is common in secondary-school and first-year university syllabi. It links to neighbouring topics Factorial and binomial topics, Multivariate discrete distributions, so understanding it makes those chapters shorter.
In everyday life: Look for Negative multinomial 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 Negative multinomial distribution in 20 minutes

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

Frequently asked questions

What is Negative multinomial distribution in simple terms?

In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(x0, p)) to more than two outcomes. As with the univariate negative binomial distribution, if the parameter x 0 {\displaystyle x_{0}} is a positive integer, the n…

Why does Negative multinomial 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 Negative multinomial 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 Negative multinomial distribution.