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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 expect
by 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 agoMGFs do not determine distributions (The Student's t distribution does not have an MGF). CDFs and characteristic functions determine distributions.
- adenadel 6y agoWell, 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 agoI'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.