Combining random variables

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Random variables are neither "random" nor "variable". However, by defining arithmetic operations on them, we can put them into equations, where they can act like variables.

If [math]X [/math] and [math]Y [/math] are random variables on a probability space [math](\href{/cs2800/wiki/index.php/S}{S},\href{/cs2800/wiki/index.php/Pr}{Pr}) [/math], then [math]X \href{/cs2800/wiki/index.php?title=%2B&action=edit&redlink=1}{+} Y [/math] is the random variable on [math](\href{/cs2800/wiki/index.php/S}{S},\href{/cs2800/wiki/index.php/Pr}{Pr}) [/math] given by [math](X \href{/cs2800/wiki/index.php?title=%2B&action=edit&redlink=1}{+} Y)(s) := X(s) + Y(s) [/math].

Note: You cannot add random variables on different sample spaces.

Similarly, we can define other operations:

If [math]X [/math] and [math]Y [/math] are random variables on a probability space [math](\href{/cs2800/wiki/index.php/S}{S},\href{/cs2800/wiki/index.php/Pr}{Pr}) [/math], then [math]X \href{/cs2800/wiki/index.php?title=%C2%B7&action=edit&redlink=1}{·} Y [/math] is the random variable on [math](\href{/cs2800/wiki/index.php/S}{S},\href{/cs2800/wiki/index.php/Pr}{Pr}) [/math] given by [math](X \href{/cs2800/wiki/index.php?title=%C2%B7&action=edit&redlink=1}{·} Y)(s) \href{/cs2800/wiki/index.php/Definition}{:=} X(s)·Y(s) [/math].

Note: You cannot multiply random variables on different sample spaces.

If [math]X [/math] is a random variable on a probability space [math](\href{/cs2800/wiki/index.php/S}{S},\href{/cs2800/wiki/index.php/Pr}{Pr}) [/math], then [math]\href{/cs2800/wiki/index.php?title=-&action=edit&redlink=1}{-} X [/math] is the random variable on [math](\href{/cs2800/wiki/index.php/S}{S},\href{/cs2800/wiki/index.php/Pr}{Pr}) [/math] given by [math](\href{/cs2800/wiki/index.php?title=-&action=edit&redlink=1}{-} X)(s) := -X(s) [/math].


As usual, [math]X \href{/cs2800/wiki/index.php?title=-&action=edit&redlink=1}{-} Y [/math] is shorthand for [math]X \href{/cs2800/wiki/index.php?title=%2B&action=edit&redlink=1}{+} (\href{/cs2800/wiki/index.php?title=-&action=edit&redlink=1}{-}Y) [/math].

For example, suppose we modeled an experiment where we randomly selected a rectangle from a given set. We might have random variables [math]W [/math] and [math]H [/math] that give the width and height of the selected rectangle. We could then define a new "area" random variable by multiplying [math]W [/math] and [math]H [/math]; this would work as expected: to find the area of a given outcome, you would measure the width and the height and then multiply them (since by definition, [math]A(s) = (W·H)(s) = W(s)H(s) [/math]).

Because we define operations on random variables pointwise, random variables behave the same way as real numbers do. For example,


If [math]X [/math], [math]Y [/math], and [math]Z [/math] are random variables on a probability measure [math](\href{/cs2800/wiki/index.php/S}{S},\href{/cs2800/wiki/index.php/Pr}{Pr}) [/math], then [math]X(Y + Z) = XY + XZ [/math].
Proof:
Choose an arbitrary [math]s \href{/cs2800/wiki/index.php/%E2%88%88}{∈} \href{/cs2800/wiki/index.php/S}{S} [/math]. We have

[math]\begin{align*} \left(X(Y + Z)\right)(s) &= X(s)\left(Y+Z\right)(s) && \href{/cs2800/wiki/index.php/%C2%B7_(random_variables)}{\text{by definition of ·}} \\ &= X(s)\left(Y(s) + Z(s)\right) && \href{/cs2800/wiki/index.php/%2B_(random_variables)}{\text{by definition of +}} \\ &= X(s)Y(s) + X(s)Z(s) && \href{/cs2800/wiki/index.php/Arithmetic}{arithmetic} \\ &= (XY)(s) + (XZ)(s) && \href{/cs2800/wiki/index.php/%C2%B7_(random_variables)}{\text{by definition of ·}} \\ &= (XY + XZ)(s) && \href{/cs2800/wiki/index.php/%2B_(random_variables)}{\text{by definition of +}} \\ \end{align*} [/math]

Thus [math]X(Y+Z) \href{/cs2800/wiki/index.php/Equality_(functions)}{=} XY + XZ [/math].