Give an example of two discrete random variables X and Y on...

Give an example of two discrete random variables $X$ and $Y$ on the same sample space such that $X$ and $Y$ have the same distribution, with support $\{1,2, \ldots, 10\}$, but the event $X=Y$ never occurs. If $X$ and $Y$ are independent, is it still possible to construct such an example?

Key Concepts

Discrete Random Variables

Discrete random variables map outcomes of a probability experiment to a countable set of numbers. They are defined by their probability mass functions, which assign probabilities to each individual outcome within the support. Understanding discrete random variables is essential when constructing examples or proofs in probability theory.

Support of a Random Variable

The support of a random variable is the set of values where the variable has positive probability. In this context, both X and Y having support {1, 2, ..., 10} means that they only take these values with non-zero probability. Recognizing the support is crucial when designing and analyzing probability distributions.

Identical (Marginal) Distributions

When two random variables have identical or the same marginal distributions, they assign the same probabilities to each of the values in their respective supports. This concept allows us to compare and couple variables while maintaining the consistency of their individual probabilistic behaviors.

Dependent vs. Independent Random Variables

Dependent random variables are linked through some defined relationship or joint distribution, which can be manipulated to achieve properties such as X never equaling Y. In contrast, independent random variables have a joint distribution that factors into the product of their marginal distributions. This independence constraint means that events are determined solely by their individual probabilities, making it impossible to force a condition like X ? Y for all outcomes when the variables have the same distribution.

Joint Distribution Construction

The joint distribution of two random variables defines the probability associated with each pair of outcomes. By carefully designing the joint distribution, one can enforce specific relationships between variables, such as ensuring they never take the same value. This construction is straightforward when the variables are dependent, but becomes unfeasible under the independency constraint given the shared support.

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