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> The random variable X is a function mapping the outcome to its probability This is simply wrong. Random variables are not defined this way.

by blt 2y ago

defrost 2y ago
It would be more constructive if you were to provide an example definition that you consider correct and perhaps even a link to where it is defined and used as you have yet to actually say.
- blt 2y ago
  The standard definition is the one given in the two bullet points at the top of the article we are discussing. Or, in more detail, at https://en.wikipedia.org/wiki/Random_variable#Definition https://en.wikipedia.org/wiki/Random_variable#Definition.
  - defrost 2y ago
    glitchc stated: > The random variable X is a function mapping the outcome to its probability. you've stated: "The standard definition is the one given in the two bullet points at the top of the article we are discussing" ie: > The random variable X itself, that is, the function which maps sample space outcomes to numbers. you've also stated: "Or, in more detail, at (wikipedia link)" which has: > A random variable is a measurable function from a sample space as a set of possible outcomes to a measurable space These all appear to be in rough alignment .. all three agree upon a function mapping from outcomes to measure. You've described the first as "This is simply wrong. Random variables are not defined this way." Perhaps you can expand on why this is so wrong compared to the other two definitions.
    - blt 2y ago
      In the bullet point definition, > the function which maps sample space outcomes to numbers, the "numbers" referred to in the codomain are not probabilities. Similarly, in the Wikipedia definition, > a measurable function from a sample space... to a measurable space, the "measurable space" codomain has nothing to do with probability of events. Random variables are related to probability distributions/measures, but they are not the same kind of mathematical object. As a concrete example, consider gambling on a fair coin flip. We can bet 1 unit of money on Heads and define a random variable for the amount of money we win/lose. First we set up the probability space. The sample space is the 2-element set of coin sides, Ω = {H, T} (Heads and Tails). The event set (aka sigma-algebra) F is the power set of Ω: F = {{}, {H}, {T}, {H, T}}. The probability measure assigns probability 0 to the event {}, probability 0.5 to each of the events {H}, {T}, and probability 1 to the event {H, T}. Now, we can define the random variable of our winnings by the function X mapping Ω to the real numbers by Χ(ω) = 1 if ω = H, -1 if ω = T. So, within the setting of the probability space (Ω, F, P), the random variable is defined by the function X, which does not map anything to a probability. It maps coin sides to real numbers, including negative ones! We have not yet constructed any object that gives the probabilities of the possible values of X. The only thing that "maps an outcome to a probability" in our setup so far is the measure P, but that is defined independently of our numeric random variable. If we want to talk about the probabilities that X takes the values -1 and 1, we are talking about the pushforward measure of P by X. This is another mathematical object, distinct from the random variable X itself. This is one of the points of the article. Suppose we define another random variable Y for the winnings of the person we are gambling against: Y(ω) = -1 if ω = H, 1 if ω = T. Now X and Y are two different random variables, but they have the same distributions! The pushforward measures of P by X and Y both put probability 0.5 on the measurable sets {-1} and {1}.
      - defrost 2y ago
        So your quibble boils down to the lack of a shim that normalises the mapping to numbers (or a measure) to a probabilty? Is that a massive assumption that such a thing exists that makes the GP comment egregiously wrong, or more of an impatient jump to the end as such a thing always (?) exists?
    - kgwgk 2y ago
      > These all appear to be in rough alignment Do they? What about? “The random variable X is a function mapping each outcome to the age of one of my cousins”? Does that seem like a valid definition of the concept of random variable to you because the age of my cousins are numbers - just like probabilities mentioned by glitchc’s comment are numbers?
      - defrost 2y ago
        Leaving aside that I'm mainly drawing out a proper expansion of an opaque non constructive comment from the PoV of another; "probabilities" and "age of one of my cousins" do not sound at all similiar. One has the feel of a potentially continuous interval of values that can be mapped | scaled to perhaps match up with other forms of expression, the other is a finite discrete set of probably integers between zero and 120 with total set size likely less than a 100.
        kgwgk 2y ago
        Ok. Consider the age in days or microseconds of my cousins if that makes it sound more similar. The point is that the random variable is about mappings to general numbers. “The random variable X is a function mapping the outcome to its probability” makes as much sense as “The random variable X is a function mapping the outcome to the exponential of the square root of its probability” Those are valid definitions of two different random variables - not a general definition of “random variable”.

> The random variable X is a function mapping the outcome to its probability This is simply wrong. Random variables are not defined this way.

by blt 2y ago

> The random variable X is a function mapping the outcome to its probability This is simply wrong. Random variables are not defined this way.

defrost 2y ago

It would be more constructive if you were to provide an example definition that you consider correct and perhaps even a link to where it is defined and used as you have yet to actually say.

blt 2y ago
The standard definition is the one given in the two bullet points at the top of the article we are discussing. Or, in more detail, at https://en.wikipedia.org/wiki/Random_variable#Definition https://en.wikipedia.org/wiki/Random_variable#Definition.
- defrost 2y ago
  glitchc stated: > The random variable X is a function mapping the outcome to its probability. you've stated: "The standard definition is the one given in the two bullet points at the top of the article we are discussing" ie: > The random variable X itself, that is, the function which maps sample space outcomes to numbers. you've also stated: "Or, in more detail, at (wikipedia link)" which has: > A random variable is a measurable function from a sample space as a set of possible outcomes to a measurable space These all appear to be in rough alignment .. all three agree upon a function mapping from outcomes to measure. You've described the first as "This is simply wrong. Random variables are not defined this way." Perhaps you can expand on why this is so wrong compared to the other two definitions.
  - blt 2y ago
    In the bullet point definition, > the function which maps sample space outcomes to numbers, the "numbers" referred to in the codomain are not probabilities. Similarly, in the Wikipedia definition, > a measurable function from a sample space... to a measurable space, the "measurable space" codomain has nothing to do with probability of events. Random variables are related to probability distributions/measures, but they are not the same kind of mathematical object. As a concrete example, consider gambling on a fair coin flip. We can bet 1 unit of money on Heads and define a random variable for the amount of money we win/lose. First we set up the probability space. The sample space is the 2-element set of coin sides, Ω = {H, T} (Heads and Tails). The event set (aka sigma-algebra) F is the power set of Ω: F = {{}, {H}, {T}, {H, T}}. The probability measure assigns probability 0 to the event {}, probability 0.5 to each of the events {H}, {T}, and probability 1 to the event {H, T}. Now, we can define the random variable of our winnings by the function X mapping Ω to the real numbers by Χ(ω) = 1 if ω = H, -1 if ω = T. So, within the setting of the probability space (Ω, F, P), the random variable is defined by the function X, which does not map anything to a probability. It maps coin sides to real numbers, including negative ones! We have not yet constructed any object that gives the probabilities of the possible values of X. The only thing that "maps an outcome to a probability" in our setup so far is the measure P, but that is defined independently of our numeric random variable. If we want to talk about the probabilities that X takes the values -1 and 1, we are talking about the pushforward measure of P by X. This is another mathematical object, distinct from the random variable X itself. This is one of the points of the article. Suppose we define another random variable Y for the winnings of the person we are gambling against: Y(ω) = -1 if ω = H, 1 if ω = T. Now X and Y are two different random variables, but they have the same distributions! The pushforward measures of P by X and Y both put probability 0.5 on the measurable sets {-1} and {1}.
    - defrost 2y ago
      So your quibble boils down to the lack of a shim that normalises the mapping to numbers (or a measure) to a probabilty? Is that a massive assumption that such a thing exists that makes the GP comment egregiously wrong, or more of an impatient jump to the end as such a thing always (?) exists?
  - kgwgk 2y ago
    > These all appear to be in rough alignment Do they? What about? “The random variable X is a function mapping each outcome to the age of one of my cousins”? Does that seem like a valid definition of the concept of random variable to you because the age of my cousins are numbers - just like probabilities mentioned by glitchc’s comment are numbers?
    - defrost 2y ago
      Leaving aside that I'm mainly drawing out a proper expansion of an opaque non constructive comment from the PoV of another; "probabilities" and "age of one of my cousins" do not sound at all similiar. One has the feel of a potentially continuous interval of values that can be mapped | scaled to perhaps match up with other forms of expression, the other is a finite discrete set of probably integers between zero and 120 with total set size likely less than a 100.
      - kgwgk 2y ago
        Ok. Consider the age in days or microseconds of my cousins if that makes it sound more similar. The point is that the random variable is about mappings to general numbers. “The random variable X is a function mapping the outcome to its probability” makes as much sense as “The random variable X is a function mapping the outcome to the exponential of the square root of its probability” Those are valid definitions of two different random variables - not a general definition of “random variable”.