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Original question on cross validated: https://stats.stackexchange.com/questions/628352/difference-between-random-vector-and-joint-distribution

My definition of a random vector is a vector $(x_{1},...,x_{n})$ that maps from a sample space to ${R}^{n}$. An example of this (in my understanding) would be a random process such as drawing one card from a deck. Let us say all potential suits of the card are the sample space {hearts, clubs, diamonds, spades}. An example of a random vector is a vector $(X_{1}, X_{2})$, where $X_{1}, X_{2}$ each map the same sample space to real numbers.

In slide 6 at this link it describes a joint RV that considers the outcome of a coin flip and a dice roll. In this case, the two sample spaces {H,T} and {1,2,3,4,5,6} are disjoint. Would this example not be considered a random vector, then?

My confusion is the difference between these two constructions.

There is a similar outstanding stackexchange question that discusses this but does not seem to answer it fully, so I am reposting as a separate question: what is the difference between joint probability distribution and random vector

mk0219
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I would say that, when you flip a coin and independently roll a die, the result is not a random vector, but only for a subtle reason. The components of a vector must be numbers (usually real numbers, but they are sometimes complex numbers, or elements of a finite field). However, "heads" and "tails" are not numbers, and that is what the first component of your outcome space is.

You are correct to notice that the two constructions are quite similar. Both are choosing a random element of a Cartesian product, so a random ordered pair. But if you call something a vector, you should be able to add it to other vectors, and multiply it by a scalar, and that is just not possible with "heads" and "tails."

Mike Earnest
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