# Probability that X is less than a set

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## Main Question or Discussion Point

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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andrewkirk
Homework Helper
Gold Member
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.

mathman
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.

If there is more than one $x$ for which $g(x)=y$ then $g^{-1}(y)$ is a set.

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?