# Probability that X is less than a set

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• showzen
In summary, the textbook states that the cdf of ##Y=g(X)## is equal to the set of all numbers less than or equal to the given number, ##g^{-1}(y)##. However, this last equality is confusing and it is not clear what is being defined by ##X##.

#### showzen

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?

showzen said:
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.

showzen said:
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.

andrewkirk said:
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?