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For a random variable $X$, we consider $X\textbf{1}_A$, where $A$ is some event. For some $x>0$, it seems that \begin{align}P(X\textbf{1}_A<x)&=P(X\textbf{1}_A<x, A\text{ is true})+P\left(X\mathbf{1}_A<x, A\text{ is not true}\right)\\[0.2cm]&=P(X<x)+P(0<x)\\[0.2cm]&=P(X<x)+1\end{align}

It is obviously wrong since any probability can not greater than 1. However, I don't know where I am wrong.

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1 Answer 1

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$X\cdot 1_A$ is a random variable which is equal to $X$ when the event $A$ occurs, and is equal to zero otherwise. Therefore if $x>0$ then $$ \mathbb{P}(X\cdot 1_A<x)=\mathbb{P}(\{X<x\}\cap A)+\mathbb{P}(A^c) $$

This probability is at most $\mathbb{P}(A)+\mathbb{P}(A^c)=1$, as expected.

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