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Maybe this question is a little bit strange, but actually, I can't handle it. Suppose we have a random variable $Z$ which is equal to random variable $X$ with probability $p$ and is equal to random variable $Y$ with probability $q$. Can i write $Z=X\times p + Y \times q$? Otherwise how should i write it?

Optimized Life
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    You could write $\pi_Z=p\pi_X+q\pi_Y$, where $\pi_Z$ is the probability law of $Z$, etc, and say "the law of $Z$ is a $p,q$ mixture of the laws of $X$ and $Y$". Or you could introduce a new Bernoulli rv $B$ (independent of $X$ and $y$ with $P(B=1)=p$ and write $Z=BX+(1-B)Y$. – kimchi lover Mar 08 '22 at 18:40
  • Thanks a lot for your clear response – Optimized Life Mar 08 '22 at 19:22
  • If $X$ and $Y$ are defined on a probability space $(\Omega,\mathcal F,\mathbb P)$, one could write $Z= X\cdot\mathsf 1_E + Y\cdot(1-\mathsf 1_E)$ where $\in\mathcal F$ such that $\mathbb P(E)=p$. – Math1000 Mar 21 '22 at 03:27

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