Probability that X is less than a set

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Hi everyone, I am currently working through the textbook Statistical Inference by Casella and Berger. My question has to do with transformations.

Let ##X## be a random variable with cdf ##F_X(x)##. We want to find the cdf of ##Y=g(X)##. So we define the inverse mapping, ##g^{-1}(\{y\})=\{x\in S_x | g(x)=y\}##. Now, ##F_Y(y)=P(Y\leq y)=P(g(X)\leq y)=P(X\leq g^{-1}(y))##.

My textbook then states ##P(X\leq g^{-1}(y))=P(\{x\in S_x | g(x)\leq y\}##.

The issue I have is with this last equality. Are we defining the meaning of ##X## less than or equal to a set here, or am I missing some intuition on sets?
 

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  • #2
andrewkirk
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Are we defining the meaning of XX less than or equal to a set here, or am I missing some intuition on sets?
It looks like the former. You are not missing any intuition on sets. It is non-standard notation, which I have not encountered before. But if you have to use that book then you'll need to bear with their notation.
 
  • #3
mathman
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Hi everyone, I am currently working through the textbook Statistical Inference by Casella and Berger. My question has to do with transformations.

Let ##X## be a random variable with cdf ##F_X(x)##. We want to find the cdf of ##Y=g(X)##. So we define the inverse mapping, ##g^{-1}(\{y\})=\{x\in S_x | g(x)=y\}##. Now, ##F_Y(y)=P(Y\leq y)=P(g(X)\leq y)=P(X\leq g^{-1}(y))##.

My textbook then states ##P(X\leq g^{-1}(y))=P(\{x\in S_x | g(x)\leq y\}##.

The issue I have is with this last equality. Are we defining the meaning of ##X## less than or equal to a set here, or am I missing some intuition on sets?
The notation is a little confusing. ##g^{-1}(y)## is a number, ##g^{-1}(\{y\})## is a set.
 
  • #4
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If there is more than one ##x## for which ##g(x)=y## then ##g^{-1}(y)## is a set.
 
  • #5
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It looks like the former. You are not missing any intuition on sets. It is non-standard notation, which I have not encountered before. But if you have to use that book then you'll need to bear with their notation.
Is there any supplementary resource that you would recommend?
 

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