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Let $\alpha$ and $\beta$ be independent random variables uniformly distributed from $0$ to $1$.
Let $\lambda= k_1\alpha -c_1\alpha + k_2\beta -c_2\beta$.
Let $x$ be the random variable that is uniformly between distributed from $-\lambda$ to $\lambda$.

How do I split $x$ into two random variable ($y$ and $z$), one with the $k_i$ constants and one with the $c_i$ constants such that $x=y+z$?

In essence $x$ is a random variable over a random interval. Is there a simply way to split the random variables?

If not, is there a way of splitting $x$ such that $E[x]=E[y]+E[z]$ with y with the $k_i$ constants and x with the $c_i$ constants.

tinyhippo
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  • Your last question is too permissive: take $y$ to be any random variable whatsoever, and define $z-x=y$. Also, is your intention that $\alpha$ and $\beta$ be independent random variables? – Greg Martin Mar 02 '15 at 18:19
  • I see your point. I have updated the question to clarify. – tinyhippo Mar 03 '15 at 09:47

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