# Equivalent definitions of random variable

• I
• mnb96
In summary, the conversation discusses the definition of a random variable and its relationship to a cumulative distribution function. It is clarified that a random variable cannot be defined solely by its associated CDF, as two different random variables can have the same CDF but be defined on different probability spaces. The need for a graduate course in probability to fully understand this concept is also mentioned.
mnb96
Hello,

According to the Wikipedia article on random variables:
"Any random variable can be described by its cumulative distribution function, which describes the probability that the random variable will be less than or equal to a certain value."
If the above statement is true, then, instead of defining a (real) random variable as a function from a sample space of some probability space to the reals, could we equivalently define it as a subset of ℝ associated with a CDF?

No. To one and the same CDF, there might be multiple very different associated random variables. Sure, these random variables will have the same probability distribution, but they're not the same as function.

FactChecker
mnb96 said:

In its current state, that article has some self-contradictions. The abstract definition it gives for random variable looks correct. The statements it makes about cumulative distributions are misleading.

Consider the following situation. We have two questionnaires. Questionnaire A is designed to rate a persons interest in applied mathematics "on a scale" from 0 to 10. Questionnaire B is designed to rate a person's interest in basketball on a scale from 0 to 10.

A realization of Random variable X is defined as "Pick a student at your school at random and administer questionnaire A. Let X be the student's score on the questionnaire. A realization of Random variable Y is defined as "Pick a student at you school at random and administer questionnaire B. Let Y be the student's score on questionnaire B.

It is possible that random variable X and random variable Y might have the same distribution and same cumulative distribution. But the two random variables are not the same random variable because they are not defined on the same probability space. X is defined on a space of events having to do with questionnaire A while Y is defined on a space of events having to do with questionnaire B.

mnb96 and MarneMath
Hey mnb96.

A graduate course in probability will make this definition more precise.

The idea is that you have a sigma algebra, a Borel space and you mix the two up to make the idea of a probability space [with its events and actual probabilities] consistent with what a probability space actually is.

OK.
Thank you all. I think all the answers I received were pretty clear.

## 1. What is a random variable?

A random variable is a variable whose value is determined by the outcome of a random event. It is often denoted by a capital letter, such as X or Y, and can take on different numerical values based on the probability of different outcomes.

## 2. What are equivalent definitions of a random variable?

There are two main equivalent definitions of a random variable. The first is a function that maps each outcome of a random event to a numerical value. The second is a probability distribution that assigns probabilities to each possible value of the random variable.

## 3. How are random variables related to probability distributions?

Random variables and probability distributions are closely related. A random variable is defined by a probability distribution, which describes the likelihood of each possible value of the variable. In turn, the properties of a random variable can be used to calculate probabilities of events related to the random variable.

## 4. What is the difference between a discrete and a continuous random variable?

A discrete random variable can only take on a finite or countably infinite number of values, while a continuous random variable can take on any value within a certain range. Discrete random variables are often associated with events that can be counted, while continuous random variables are associated with measurements or quantities.

## 5. How do you determine the probability of a random variable?

The probability of a random variable can be determined by using its probability distribution. The probability of a specific value of the random variable is given by the corresponding probability in the distribution. The probability of an event related to the random variable, such as the sum of two random variables, can be calculated using mathematical properties and rules.

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