Pairwise independence

Key takeaways

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

Reference excerpt

In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent. Any collection of mutually independent random variables is pairwise independent, but some pairwise independent collections are not mutually independent. Pairwise independent random variables with finite variance are uncorrelated. A pair of random variables X and Y are independent if and only if the random vector (X, Y) with joint cumulative distribution function (CDF) F X , Y ( x , y ) {\displaystyle F_{X,Y}(x,y)} satisfies

F X , Y ( x , y ) = F X ( x ) F Y ( y ) , {\displaystyle F_{X,Y}(x,y)=F_{X}(x)F_{Y}(y),}

or equivalently, their joint density f X , Y ( x , y ) {\displaystyle f_{X,Y}(x,y)} satisfies

f X , Y ( x , y ) = f X ( x ) f Y ( y ) . {\displaystyle f_{X,Y}(x,y)=f_{X}(x)f_{Y}(y).}

That is, the joint distribution is equal to the product of the marginal distributions. Unless it is not clear in context, in practice the modifier "mutual" is usually dropped so that independence means mutual independence. A statement such as " X, Y, Z are independent random variables" means that X, Y, Z are mutually independent.

Example Pairwise independence does not imply mutual independence, as shown by the following example attributed to S. Bernstein. Suppose X and Y are two independent tosses of a fair coin, where we designate 1 for heads and 0 for tails. Let the third random variable Z be equal to 1 if exactly one of those coin tosses resulted in "heads", and 0 otherwise (i.e., Z = X ⊕ Y {\displaystyle Z=X\oplus Y} ). Then jointly the triple (X, Y, Z) has the following probability distribution:

( X , Y , Z ) = { ( 0 , 0 , 0 ) with probability 1 / 4 , ( 0 , 1 , 1 ) with probability 1 / 4 , ( 1 , 0 , 1 ) with probability 1 / 4 , ( 1 , 1 , 0 ) with probability 1 / 4. {\displaystyle (X,Y,Z)={\begin{cases}(0,0,0)&{\text{with probability}}\ 1/4,\\(0,1,1)&{\text{with probability}}\ 1/4,\\(1,0,1)&{\text{with probability}}\ 1/4,\\(1,1,0)&{\text{with probability}}\ 1/4.\end{cases}}}

Here the marginal probability distributions are identical: f X ( 0 ) = f Y ( 0 ) = f Z ( 0 ) = 1 / 2 , {\displaystyle f_{X}(0)=f_{Y}(0)=f_{Z}(0)=1/2,} and

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Pairwise independence

Start with the simplest possible case. Write down what Pairwise independence 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 Pairwise independence 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 Pairwise independence 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 Pairwise independence

In research: Pairwise independence 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 Pairwise independence 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: Pairwise independence is common in secondary-school and first-year university syllabi. It links to neighbouring topics Independence (probability theory), Theory of probability distributions, so understanding it makes those chapters shorter.
In everyday life: Look for Pairwise independence 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 Pairwise independence in 20 minutes

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

Frequently asked questions

What is Pairwise independence in simple terms?

In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent. Any collection of mutually independent random variables is pairwise independent, but some pairwise independent collections are not mutually independent.

Why does Pairwise independence 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 Pairwise independence?

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 Pairwise independence.