Probability that X is less than a set

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##.
  • #1
showzen
34
0
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
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.
 
  • #3
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.
 
  • #4
If there is more than one ##x## for which ##g(x)=y## then ##g^{-1}(y)## is a set.
 
  • #5
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?
 

1. What is the meaning of "Probability that X is less than a set"?

The probability that X is less than a set refers to the likelihood that a value of X will fall below a certain threshold or cutoff point. This is often used in statistical analysis to determine the chance of a particular event occurring.

2. How is the probability that X is less than a set calculated?

The calculation of the probability that X is less than a set depends on the type of distribution and the specific values being considered. For continuous distributions, it involves finding the area under the curve of the distribution up to the desired value. For discrete distributions, it involves adding up the individual probabilities of all values less than the set.

3. Can the probability that X is less than a set be greater than 1?

No, the probability that X is less than a set cannot be greater than 1. This is because probability is a measure of likelihood and ranges from 0 to 1, where 0 means impossible and 1 means certain.

4. How does the probability that X is less than a set relate to the mean and standard deviation of a distribution?

The probability that X is less than a set is affected by the mean and standard deviation of a distribution. Generally, as the mean increases, the probability of X being less than a set also increases. And as the standard deviation increases, the probability of X being less than a set decreases.

5. How can the probability that X is less than a set be used in real-life situations?

The probability that X is less than a set can be used in various real-life situations, such as in finance to determine the likelihood of stock prices falling below a certain level, in medicine to assess the chance of a treatment being effective, and in sports to predict the probability of a team winning a game based on their past performance.

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