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Math Help - Multivariate random variable

  1. #1
    Member courteous's Avatar
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    Question Multivariate random variable

    Let X and Y be random variables and a,b,c,d \in \mathbb{R}. When is
    Z=\left[ \begin{matrix}a & b \\ c & d \end{matrix} \right]\left[ \begin{matrix}X \\ Y\end{matrix} \right]
    a multivariate random variable?

    1. when aX+bY\neq cX+dY
    2. always
    3. when ad-bc\neq 0
    4. when aX+bY=cX+dY
    5. never
    6. when ad-bc=1
    Why is 2nd answer the right one? Just because Z=\left[ \begin{matrix}aX+bY \\ cX+dY \end{matrix} \right] is always a "well defined" (I'm making the terminology up) and, after all, that is the definition of a multivariate random variable (Wikipedia doesn't give one, though)?
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  2. #2
    Moo
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    Because aX+bY and cX+dY are random variables and that makes Z being a multivariate random variable (a vector which components are random variables).
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