Timeline for Basic question about labeling of random variables apropos of the definition of convexity
Current License: CC BY-SA 4.0
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| Mar 6, 2020 at 0:26 | vote | accept | Antoni Parellada | ||
| Mar 5, 2020 at 18:02 | history | edited | whuber♦ |
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| Mar 5, 2020 at 15:27 | answer | added | whuber♦ | timeline score: 2 | |
| Mar 5, 2020 at 14:58 | history | edited | Antoni Parellada | CC BY-SA 4.0 |
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| Mar 5, 2020 at 14:46 | history | edited | Antoni Parellada | CC BY-SA 4.0 |
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| Mar 5, 2020 at 14:27 | comment | added | Antoni Parellada | @whuber Here is where I get stuck... We wouldn't refer to the realization $x_5=5$ of the random variable generated by throwing a dice with probability $w_5=1/6$ as a random variable, but as a possible value of the random variable. It has to be embarrassingly obvious, but I would like to make sure I get it right conceptually, not just intuitively or computationally. | |
| Mar 5, 2020 at 14:17 | comment | added | whuber♦ | Yes, indexing works here exactly the way it works everywhere else: each distinct $i$ designates a random variable. Taleb's expression is merely a consequence of the definition of convexity you quote and is easily demonstrated inductively: a linear combination of three values is a linear combination of two values, of which one is itself a linear combination, etc. | |
| Mar 5, 2020 at 14:05 | history | edited | Antoni Parellada | CC BY-SA 4.0 |
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| Mar 5, 2020 at 14:03 | comment | added | Antoni Parellada | @whuber Thank you very much for your comment. I thought what I was quoting was the definition of convexity in the context of random variables. A vector of probabilities adding up to 1 makes a lot of sense, as when trying to find the expectation of a discrete random variable, but I find the index $i$ rather confusing... If it denotes each one of the fractions adding up to 1, does $X_i$ denote the specific values the random variable can take, more typically written as $x_i,$ as in $X=\{1,\dots,6\}$ in for instance $x_5=5$ in a dice? | |
| Mar 5, 2020 at 13:43 | comment | added | whuber♦ | You are not quoting a definition of random variables, Antoni: that expression is merely a consequence of the assumed convexity of the function $f.$ The author has (rather blithely, IMO) assumed the reader understands the $w_i$ are non-negative real numbers that sum to unity. | |
| Mar 5, 2020 at 13:10 | history | asked | Antoni Parellada | CC BY-SA 4.0 |