DenisAmbrosov
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What does it mean: 'The inf is only needed when measuring the quantile of distributions that have regions with no probability'?
Chapter 2 Random Variables
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Hi @DenisAmbrosov Inf refers to "Infimum" (see https://en.wikipedia.org/wiki/Infimum_and_supremum) and is a more robust (but effectively still a) minimum:
The classic counter-example is a open-ended uniform distribution. Please note our typical uniform is closed per https://en.wikipedia.org/wiki/Continuous_uniform_distribution, so the uniform's support is [0,1] and is closed because it specifically includes the zero (and the one). But if it were (0,1] and did not include the zero (i.e., open on the zero), then this function would not have a minimum; if you said it is X, then we can counter with X/2. Whatever you say is the minimum, we can divide by 2 and get a smaller number (the arrow never reaches the target!). But the infimum of (0,1] is zero, so the infimum is here a "more robust minimum" or a most robust greatest lower bound. Thanks,
I really don't know where it matters, realistically. This quantile is our value at risk (VaR) definition. For example, if the distribution is a standard normal and we want the 5.0% quantile (i.e., α = 0.050), the quantile is =NORM.S.INV(0.050) = -1.644853627 is an Infinium because that is the greatest lower bound (GLB); e.g., the 5.0% quantile is not the values of -1.640 or -1.630 which are lower. This quantile here is both the minimum and the infimum and we don't typically need an infimum because the quantile is in the set; i.e., our quantile -1.644853627 is in the set of real numbers that support [aka, domain] our distribution.
The classic counter-example is a open-ended uniform distribution. Please note our typical uniform is closed per https://en.wikipedia.org/wiki/Continuous_uniform_distribution, so the uniform's support is [0,1] and is closed because it specifically includes the zero (and the one). But if it were (0,1] and did not include the zero (i.e., open on the zero), then this function would not have a minimum; if you said it is X, then we can counter with X/2. Whatever you say is the minimum, we can divide by 2 and get a smaller number (the arrow never reaches the target!). But the infimum of (0,1] is zero, so the infimum is here a "more robust minimum" or a most robust greatest lower bound. Thanks,
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