Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

In my understanding, random variable is a measurable function from a probability space to a measurable space. Suppose $X$ is a random variable from $(A, \sigma_{A},P_A)$ to $(B,\sigma_{B})$. And $Y$ is a random variable from $(B, \sigma_{B}, P_B)$ to $(C, \sigma_{C})$.

Then $Y(X)$ is a random variable from $A$ to $C$. The problem is here, in the space $B$, we can induce probability from $A$ by random variable $X$, s.t., $P_{B}'(I) = P_A(X^{-1}(I))$. Also, random variable $Y$ use the probability measure $P_B$, then if we want to calculate the probability distribution of $Y(X)$, what probability should be used, $P_B'$ or $P_B$, or both?

share|improve this question
add comment

1 Answer

up vote 3 down vote accepted

$Y(X)$ doesn't mean anything. You can't define the composition of random variables. What you can do is compose a random variable $X$ by a measurable function $f$ (provided the $\sigma$ algebras are the same) : $f(X)$.

So in your example, there are two different objects, measurable functions and random variables :

-the measurable functions $f$ from $(A,\sigma_A)$ to $(B,\sigma_B)$ and $g$ from $(B,\sigma_B)$ to $(C,\sigma_C)$. Since B uses the same $\sigma$-algebra, the function $g \circ f$ is measurable from $(A,\sigma_A)$ to $(C,\sigma_C)$.

-the random variable when you add a probability distribution to the measurable spaces. So if you add $P_A$ to $(A,\sigma_A)$, the measurable function $f$ from $(A,\sigma_A)$ to $(B,\sigma_B)$ induces a random variable we can write $X$. Now since we also have a measurable function $g \circ f$ from $(A,\sigma_A)$ to $(C,\sigma_C)$, it also induces another random variable that we can write $X'$ or more usually $g(X)$. And if you add $P_B$ to $(B,\sigma_B)$, function $g$ induces a random variable we'll write $Y$.

But the composition $Y(X)$ doesn't makes any sense.

share|improve this answer
    
Thank you for your answer. Perhaps, I shouldn't associate probability to a random variable but define probability measure for some $sigma$-algebra instead. If I define random variable $Y$ in $(B, \sigma_B)$, I should use the probability induced by $X$ when I intend to use the form like $Y(X)$. BTW, when $X$ is the parameters in $Y$'s distribution function, it would be fine to define another probability measure on $(B, \sigma_B)$ for $Y$. –  itsuper7 Sep 17 '12 at 14:08
    
Well, you could define a random variable this way and write it $X$ or $Y$, but this is already defined as a measurable function, and it is usually written with small letters life $f$ or $g$ :) So if you want to be understood, you should speak of $g \circ f$ (measurable function), or $g(X)$ for the random variable, but not $Y(X)$. –  Adrien Sep 19 '12 at 8:36
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.