(a) Let X and Y be numbers that are chosen independently and...

(a) Let $X$ and $Y$ be numbers that are chosen independently and uniformly at random from $\{0,1, \ldots, n\}$. Let $Z$ be their sum modulo $n+1$. Show that $X, Y$, and $Z$ are pairwise independent but not independent. (b) Extend this example to give a collection of random variables that are $k$-wise independent but not $(k+1)$-wise independent.

(a) Let $X$ and $Y$ be numbers that are chosen independently and uniformly at random from $\{0,1, \ldots, n\}$. Let $Z$ be their sum modulo $n+1$. Show that $X, Y$, and $Z$ are pairwise independent but not independent.
(b) Extend this example to give a collection of random variables that are $k$-wise independent but not $(k+1)$-wise independent.

Key Concepts

Pairwise Independence

Pairwise independence means that for any two random variables taken from a collection, the joint distribution factors into the product of their individual distributions, regardless of the dependence among larger sets of variables. This property is crucial in probability theory and statistics when analyzing the interactions between pairs of variables, even if the entire set does not enjoy full mutual independence.

Mutual (or Joint) Independence

Mutual independence is a stronger condition where every subset of the collection of random variables has a joint distribution that equals the product of their individual distributions. When random variables are mutually independent, knowledge of the outcomes of any collection of them does not affect the probabilities of outcomes for the remaining ones. This concept is central to many probabilistic models and results.

Modular Arithmetic in the Construction of Random Variables

Modular arithmetic is a system of arithmetic for integers, where numbers 'wrap around' after reaching a certain value—the modulus. In constructing examples in probability, operations performed modulo an integer can enforce deterministic relationships among variables (such as a fixed sum modulo some number), thereby creating dependencies among them even if arbitrarily chosen components are pairwise independent.

k-wise Independence versus (k+1)-wise Independence

A set of random variables is said to be k-wise independent if every subset of k variables is independent, but this does not necessarily extend to subsets of k+1 or more variables. This concept is significant in computer science and combinatorics because it allows the construction of examples and models where limited independence is sufficient for certain applications, while full independence is too strong or costly to achieve.

Polynomial Methods in Finite Fields for k-wise Independence

A common method for constructing k-wise independent random variables is through the use of random polynomials over a finite field. By selecting coefficients uniformly at random for a polynomial of a fixed degree, the evaluations of this polynomial at distinct points yield random variables that are guaranteed to be k-wise independent. This technique is a powerful tool in both theoretical computer science and combinatorial design.

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