Suppose X and Y are discrete r.v.s such that P(X=Y)=1. This...

Suppose $X$ and $Y$ are discrete r.v.s such that $P(X=Y)=1$. This means that $X$ and $Y$ always take on the same value.
(a) Do $X$ and $Y$ have the same PMF?
(b) Is it possible for $X$ and $Y$ to be independent?

Key Concepts

Discrete Random Variables

Discrete random variables are functions that assign numerical values to outcomes in a discrete sample space. Their behavior is fully described by their probability mass functions (PMFs), which specify the probability that the variable takes on each possible value. Understanding their properties lays the foundation for analyzing relationships such as equality, independence, and dependence between variables.

Probability Mass Function (PMF)

The probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. It characterizes the distribution of the random variable, and when two random variables are almost surely equal (i.e., they take on the same value with probability one), they must share the same PMF because their distributions over the outcome space coincide.

Almost Sure Equality

Almost sure equality means that two random variables are equal with probability one. This concept implies that, except possibly on a set of outcomes with probability zero, the two variables take the same value. As a result, if two random variables are almost surely equal, their marginal distributions (PMFs) are identical, reflecting the same probabilities for each outcome.

Independence of Random Variables

Two random variables are independent if the occurrence of any specific value of one does not affect the probability distribution of the other, meaning that the joint probability is the product of the individual probabilities. However, when two variables are almost surely equal, knowing the value of one completely determines the value of the other; thus, unless the variables are degenerate (taking a single constant value with probability one), they cannot be independent because they exhibit perfect dependence.

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