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Very good points. The Normal Distribution has very nice properties, which adds to its popularity, which, if I understand you correctly, adds to the significance

Very good points. The Normal Distribution has very nice properties, which adds to its popularity, which, if I understand you correctly, adds to the significance we place on mean and variance.

adenadel 6y ago
I wouldn't say that the existence of the normal distribution is necessarily the reason that we place significance on the mean and variance. The mean is incredibly natural to define when you move to measure theoretic probability (simply the integral of a function, i.e. a random variable, with respect to some measure). When you take this point of view, random variables with particular moments existing are simply functions in L^p. Further, when you move on to proving the CLT the moments give you properties of the characteristic functions that allow you to prove the CLT. These are all deep connections. There's another interpretation of the mean (and conditional expectation) as quantities minimizing squared error. It's not surprising that squared error and variance are so similar and that these are connected.
- yetanothermonk 6y ago
  Great comment! You touched on something that got me wondering: do you know of something similar to MGF that determines a distribution but is not based on expected values?
  - clircle 6y ago
    MGFs do not determine distributions (The Student's t distribution does not have an MGF). CDFs and characteristic functions determine distributions.
  - adenadel 6y ago
    Well, characteristic functions are more general than MGFs (in that they are always finite) and have very useful properties, but they have a similar definition and are also based on an expectation. This may be pedantic, but an object that determines a distribution that isn't based on expected values is the CDF. In my first probability course in undergrad I think we defined probability mass functions and probability density functions first and defined CDFs in terms of them, but from a measure theoretic point of view, the CDF is more fundamental since it is defined for continuous and discrete distributions (and also for distributions that are neither).
    - contravariant 6y ago
      I'd probably choose the probability measure itself, rather than the CDF, just because you won't always have a space with a nice order topology. Of course the expectation value and the probability measure are really two sides of the same coin, the expectation value being equivalent to integration with respect to the probability measure.

Very good points. The Normal Distribution has very nice properties, which adds to its popularity, which, if I understand you correctly, adds to the significance we place on mean and variance.

adenadel 6y ago

I wouldn't say that the existence of the normal distribution is necessarily the reason that we place significance on the mean and variance. The mean is incredibly natural to define when you move to measure theoretic probability (simply the integral of a function, i.e. a random variable, with respect to some measure). When you take this point of view, random variables with particular moments existing are simply functions in L^p. Further, when you move on to proving the CLT the moments give you properties of the characteristic functions that allow you to prove the CLT. These are all deep connections. There's another interpretation of the mean (and conditional expectation) as quantities minimizing squared error. It's not surprising that squared error and variance are so similar and that these are connected.

yetanothermonk 6y ago
Great comment! You touched on something that got me wondering: do you know of something similar to MGF that determines a distribution but is not based on expected values?
- clircle 6y ago
  MGFs do not determine distributions (The Student's t distribution does not have an MGF). CDFs and characteristic functions determine distributions.
- adenadel 6y ago
  Well, characteristic functions are more general than MGFs (in that they are always finite) and have very useful properties, but they have a similar definition and are also based on an expectation. This may be pedantic, but an object that determines a distribution that isn't based on expected values is the CDF. In my first probability course in undergrad I think we defined probability mass functions and probability density functions first and defined CDFs in terms of them, but from a measure theoretic point of view, the CDF is more fundamental since it is defined for continuous and discrete distributions (and also for distributions that are neither).
  - contravariant 6y ago
    I'd probably choose the probability measure itself, rather than the CDF, just because you won't always have a space with a nice order topology. Of course the expectation value and the probability measure are really two sides of the same coin, the expectation value being equivalent to integration with respect to the probability measure.