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updated table with equations 


[SVN r50239]
This commit is contained in:
Thijs van den Berg
2008-12-11 11:32:54 +00:00
parent 21168dad49
commit 3fac5c27ee
1 changed files with 36 additions and 20 deletions
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56
doc/sf_and_dist/distributions/laplace.qbk
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@@ -41,7 +41,7 @@ probability density function:
[equation laplace_pdf]
The location and scale parameters are equivalent to the mean and
standard deviation of the logarithm of the random variable.
standard deviation.
The following graph illustrates the effect of the location
parameter on the PDF, note that the range of the random
@@ -61,11 +61,9 @@ The next graph illustrates the effect of the scale parameter on the PDF:
Constructs a laplace distribution with location /location/ and
scale /scale/.
The location parameter is the same as the mean of the logarithm of the
random variate.
The location parameter is the same as the mean of the random variate.
The scale parameter is the same as the standard deviation of the
logarithm of the random variate.
The scale parameter is proportional to the standard deviation of the random variate.
Requires that the scale parameter is greater than zero, otherwise calls
__domain_error.
@@ -83,14 +81,12 @@ Returns the /scale/ parameter of this distribution.
All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] that are generic to all
distributions are supported: __usual_accessors.
The domain of the random variable is \[0,+[infin]\].
The domain of the random variable is \[-[infin],+[infin]\].
[h4 Accuracy]
The laplace distribution is implemented in terms of the
standard library log and exp functions, plus the
[link math_toolkit.special.sf_erf.error_function error function],
and as such should have very low error rates.
standard library log and exp functions and as such should have very low error rates.
[h4 Implementation]
@@ -100,17 +96,37 @@ and /q = 1-p/.
[table
[[Function][Implementation Notes]]
[[pdf][Using the relation: pdf = e[super -(ln(x) - m)[super 2 ] \/ 2s[super 2 ] ] \/ (x * s * sqrt(2pi)) ]]
[[cdf][Using the relation: p = cdf(normal_distribtion<RealType>(m, s), log(x)) ]]
[[cdf complement][Using the relation: q = cdf(complement(normal_distribtion<RealType>(m, s), log(x))) ]]
[[quantile][Using the relation: x = exp(quantile(normal_distribtion<RealType>(m, s), p))]]
[[quantile from the complement][Using the relation: x = exp(quantile(complement(normal_distribtion<RealType>(m, s), q)))]]
[[mean][e[super m + s[super 2 ] / 2 ] ]]
[[variance][(e[super s[super 2] ] - 1) * e[super 2m + s[super 2 ] ] ]]
[[mode][e[super m + s[super 2 ] ] ]]
[[skewness][sqrt(e[super s[super 2] ] - 1) * (2 + e[super s[super 2] ]) ]]
[[kurtosis][e[super 4s[super 2] ] + 2e[super 3s[super 2] ] + 3e[super 2s[super 2] ] - 3]]
[[kurtosis excess][e[super 4s[super 2] ] + 2e[super 3s[super 2] ] + 3e[super 2s[super 2] ] - 6 ]]
[[pdf][Using the relation: pdf = e[super -abs(x-m) \/ s] \/ (2 * s) ]]
[[cdf][Using the relations:
x < m : p = e[super (x-m)/s ] \/ s
x >= m : p = 1 - e[super (m-x)/s ] \/ s
]]
[[cdf complement][Using the relation:
-x < m : q = e[super (-x-m)/s ] \/ s
-x >= m : q = 1 - e[super (m+x)/s ] \/ s
]]
[[quantile][Using the relations:
p < 0.5 : x = m + s * log(2*p)
p >= 0.5 : x = m - s * log(2-2*p)
]]
[[quantile from the complement][Using the relation:
q > 0.5: x = m + s*log(2-2*q)
q <=0.5: x = m - s*log( 2*q )
]]
[[mean][m]]
[[variance][2 * s[super 2] ]]
[[mode][m]]
[[skewness][0]]
[[kurtosis][6]]
[[kurtosis excess][3]]
]
[h4 References]