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Let $(S,\mathcal A, P)$ be a probability space and $\mathbf X:S\rightarrow \mathbb R^n$ random vector. Let $X_i:S\rightarrow \mathbb R$ be random variables such that $\mathbf X=(X_1,\ldots ,X_n)$.

Is there any difference between distribution of random vector $\mathbf X$ and joint probability distribution of random variables $X_i$?

More generally, is joint probability distribution of random variables just a different word for distribution of some random vector?

Burt Named
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  • Yes, I think there is no difference. But I am curious for a confirmation/counterargument to see if I also have a misconception on this. – Jimmy R. Mar 23 '15 at 12:58

1 Answers1

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The distribution of a random vector in $\mathbb R^n$ is exactly the same as the joint distribution of $n$ real-valued random variables. Both of these are equivalent to measures on $(\mathbb R^n,\mathcal B(\mathbb R^n))$.

S.H.W
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Mike Earnest
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