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This is a special case of a more general phenomenon: quite a lot of probability density functions (of which the exponential function is an example) can be expre
by hyperbovine 7y ago
This is a special case of a more general phenomenon: quite a lot of probability density functions (of which the exponential function is an example) can be expressed as alternating sums containing huge combinatorial terms. As if by magic, the result is always between zero and one. It’s actually a large headache for computations.
- lopmotr 7y agoYea, it's one thing to cancel out perfectly in theory but sums of numbers with big differences in magnitude aren't great for numerical stability.
- amluto 7y agoHuh? Probability density functions aren’t bounded between 0 and 1. Their integrals are exactly 1, but that’s considerably less interesting.
- anonytrary 7y agoFor example, the Dirac delta has an unbounded height and integrates to 1.
- mantap 7y agoTo be pedantic, the dirac delta is not a probability density function.
- anonytrary 7y agoThis isn't pedantic, it's irrelevant. It is a generalized density function. Whether or not it represents probability is up to the way you're using the function.
- hyperbovine 7y agoSorry, you are right, meant to say cumulative density function. (Or survival function, as in the case of exp(-x).)
- amluto 7y agoTrue, but this has never felt like math magic to me. Usually you come up with a magical function for the PDF or similar, and then normalize it so that the CDF has the right integral. That last bit is just drudgery, not magic. If you’re lucky, you can work with unnormalized distributions.