fixed "log(X) with X ~ loguniform" plot domain #59
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------- ugly proof: ------ loguniform pdf: x: [0.001, 1000] pdf = 1/(x*ln(1000/0.001)) loguniform cdf: x: [0.001, 1000] cdf value set: [0, 1] cdf = ln(x)/ln(1000/0.001) - ln(0.001)/ln(1000/0.001) inv(cdf): x: [0,1] inv(cdf) value set: [0.001, 1000] inv(cdf) = 0.001 * (1000/0.001)^x ln(inv(cdf)): x: [0, 1] ln(inv(cdf)) value set: [ln(0.001), ln(1000)] ln(inv(cdf)) = ln(0.001 * (1000/0.001)^x) inv(ln(inv(cdf))): x: [ln(0.001), ln(1000)] inv(ln(inv(cdf))) value set: [0, 1] inv(ln(inv(cdf))) = (3*ln(10) + x) / ln(1000/0.001) derivative of inv(ln(inv(cdf))): x: [ln(0.001), ln(1000)] derivative of inv(ln(inv(cdf))) = 1/ln(1000/0.001) = 0.07238241365054197127518815315277 (ln(1000) - ln(0.001)) * 0.07238241365054197127518815315277 = 1
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ugly proof:
loguniform pdf:
x: [0.001, 1000]
pdf = 1/(x*ln(1000/0.001))
loguniform cdf:
x: [0.001, 1000]
cdf value set: [0, 1]
cdf = ln(x)/ln(1000/0.001) - ln(0.001)/ln(1000/0.001)
inv(cdf):
x: [0,1]
inv(cdf) value set: [0.001, 1000]
inv(cdf) = 0.001 * (1000/0.001)^x
ln(inv(cdf)):
x: [0, 1]
ln(inv(cdf)) value set: [ln(0.001), ln(1000)]
ln(inv(cdf)) = ln(0.001 * (1000/0.001)^x)
inv(ln(inv(cdf))):
x: [ln(0.001), ln(1000)]
inv(ln(inv(cdf))) value set: [0, 1]
inv(ln(inv(cdf))) = (3*ln(10) + x) / ln(1000/0.001)
derivative of inv(ln(inv(cdf))):
x: [ln(0.001), ln(1000)]
derivative of inv(ln(inv(cdf))) = 1/ln(1000/0.001) = 0.07238241365054197127518815315277
(ln(1000) - ln(0.001)) * 0.07238241365054197127518815315277 = 1