Let $X$ and $Y$ be independent $rv$ such that $XY$ is a degenerate $rv$. Can I say that individually $X$ and $Y$ are also degenerate? Why?

# Solved – Degenerate random variable

independencejoint distributionprobabilityrandom variableself-study

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# Solved – Degenerate random variable

independencejoint distributionprobabilityrandom variableself-study

Let $X$ and $Y$ be independent $rv$ such that $XY$ is a degenerate $rv$. Can I say that individually $X$ and $Y$ are also degenerate? Why?

## Best Answer

NO. Let $X$ be any variable and $Y$ independent such that $Y=0$ with probability 1. Then $XY$ is degenerate, but $X$ need not be.This was already answered in comments:

No. Only one of them needs to be. Let $X$ be zero with probability 1 and let $Y$ be any finite-valued random variable. – Macro

To complete @Macro's thought, consider the contrapositive: when both X and Y are nondegenerate, independence implies XY must be nondegenerate. Therefore Macro has supplied all the possible counterexamples (up to permutation of X and Y). – whuber