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BetaPrimeDistribution
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BetaPrimeDistribution

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represents a beta prime distribution with shape parameters p and q.

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represents a generalized beta prime distribution with scale parameter β.

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represents a generalized beta distribution of the second kind with shape parameter α.

Details

Background & Context

  • BetaPrimeDistribution[p,q,α,β] represents a continuous statistical distribution defined over the interval and parametrized by four positive real numbers p, q, α, and β. The parameters p, q, and α are known as "shape parameters", β is known as a "scale parameter", and together, these parameters determine the overall shape of the probability density function (PDF) of the beta prime distribution. Depending on the values of p, q, α, and β, the PDF of the beta prime distribution may be unimodal or monotonic decreasing with potential singularities approaching the lower boundary of its domain. In addition, the tails of the PDF are "fat" in the sense that the PDF decreases algebraically rather than exponentially for large values . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.)
  • BetaPrimeDistribution[p,q,α,β] is sometimes referred to as the generalized beta distribution of the second kind, the inverted beta distribution, or the type VI Pearson distribution (PearsonDistribution). The two- and three-argument forms BetaPrimeDistribution[p,q] and BetaPrimeDistribution[p,q,β] evaluate to BetaPrimeDistribution[p,q,1,1] and BetaPrimeDistribution[p,q,1,β], respectively, and are sometimes referred to as the standard beta prime distribution and the generalized beta prime distribution, respectively.
  • In Bayesian analysis, the beta prime distribution arises as a prior distribution for binomial proportions expressed as odds. The beta prime distribution has also been found to model many real-world phenomena. For example, the beta prime distribution has proven useful in empirically estimating security returns and in the development of option pricing models. More recently, it has been applied to the modeling of insurance loss processes. Elsewhere, the long tail of the beta prime distribution has been shown to make the distribution particularly well suited to modeling the frequency of behaviors likely to transmit diseases among individuals versus the actual transmission of such diseases.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a beta prime distribution. Distributed[x,BetaPrimeDistribution[p,q,α,β]], written more concisely as xBetaPrimeDistribution[p,q,α,β], can be used to assert that a random variable x is distributed according to a beta prime distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
  • The probability density and cumulative distribution functions may be given using PDF[BetaPrimeDistribution[p,q,α,β],x] and CDF[BetaPrimeDistribution[p,q,α,β],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
  • DistributionFitTest can be used to test if a given dataset is consistent with a beta prime distribution, EstimatedDistribution to estimate a beta prime parametric distribution from given data, and FindDistributionParameters to fit data to a beta prime distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic beta prime distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic beta prime distribution.
  • TransformedDistribution can be used to represent a transformed beta prime distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a beta prime distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving beta prime distributions.
  • BetaPrimeDistribution is related to a number of other distributions. For example, BetaPrimeDistribution[p,q,a,b] simplifies to DagumDistribution[p,a,b] when , to SinghMaddalaDistribution[q,a,b] when , and to LogLogisticDistribution[a,b] when both and . In addition, the two-parameter form BetaPrimeDistribution[p,q] has the same PDF as the type VI Pearson distribution PearsonDistribution[6,1,f/g,1/g,1/g,0] where and and is related to both type II and type IV versions of ParetoDistribution.The PDF of BetaPrimeDistribution is a transformation of that of BetaDistribution, while the four-parameter version BetaPrimeDistribution[p,q,a,1] is the quotient of two independent random variables XGammaDistribution[p,1,a,0] and YGammaDistribution[q,1,a,0]. BetaPrimeDistribution is also related to FRatioDistribution, DirichletDistribution, KumaraswamyDistribution, NoncentralBetaDistribution, and PERTDistribution.

Examples

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Basic Examples  (12)Summary of the most common use cases

Probability density function of a beta prime distribution:

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In[1]:=1
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https://wolfram.com/xid/0e5c3ivjl42rg6-jongmz
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0e5c3ivjl42rg6-du3w70
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Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0e5c3ivjl42rg6-x2xuob
Direct link to example
Out[3]=3

Cumulative distribution function for a beta prime distribution:

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In[1]:=1
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https://wolfram.com/xid/0e5c3ivjl42rg6-irx4xy
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0e5c3ivjl42rg6-5no1wh
Direct link to example
Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0e5c3ivjl42rg6-cllkig
Direct link to example
Out[3]=3

Mean and variance of a beta prime distribution:

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In[1]:=1
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https://wolfram.com/xid/0e5c3ivjl42rg6-qpd6ob
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0e5c3ivjl42rg6-gw7f20
Direct link to example
Out[2]=2

Median of a beta prime distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-w8gy5o
Direct link to example
Out[1]=1

Probability density function of a generalized beta prime distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-4h3x87
Direct link to example
Out[1]=1
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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-5ocu4f
Direct link to example
Out[2]=2

Cumulative distribution function of a generalized beta prime distribution:

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In[1]:=1
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https://wolfram.com/xid/0e5c3ivjl42rg6-4ijf4j
Direct link to example
Out[1]=1
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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-gbiz6o
Direct link to example
Out[2]=2

Mean and variance of a generalized beta prime distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-owdtew
Direct link to example
Out[1]=1
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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-397c
Direct link to example
Out[2]=2

Median of a generalized beta prime distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-ocatsp
Direct link to example
Out[1]=1

Probability density function of a generalized beta distribution of the second kind:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-2osk2g
Direct link to example
Out[1]=1
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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-m9820l
Direct link to example
Out[2]=2

Cumulative distribution function of a generalized beta distribution of the second kind:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-lv745x
Direct link to example
Out[1]=1
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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-5lqo4o
Direct link to example
Out[2]=2

Mean and variance of a generalized beta distribution of the second kind:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-5c7ewr
Direct link to example
Out[1]=1
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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-xcqftd
Direct link to example
Out[2]=2

Median of a generalized beta distribution of the second kind:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-q61zz0
Direct link to example
Out[1]=1

Scope  (9)Survey of the scope of standard use cases

Generate a sample of pseudorandom numbers from a beta prime distribution:

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In[1]:=1
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https://wolfram.com/xid/0e5c3ivjl42rg6-nzz7to
Direct link to example

Compare its histogram to the PDF:

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In[2]:=2
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https://wolfram.com/xid/0e5c3ivjl42rg6-03mwaz
Direct link to example
Out[2]=2

Distribution parameters estimation:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-45b7g2
Direct link to example

Estimate the distribution parameters from sample data:

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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-epi747
Direct link to example
Out[2]=2

Compare the density histogram of the sample with the PDF of the estimated distribution:

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In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-f8ui5o
Direct link to example
Out[3]=3

Skewness:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-kcnlxu
Direct link to example
Out[1]=1

For a generalized beta distribution of the second kind, skewness does not depend on β:

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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-39x0zy
Direct link to example
Out[2]=2

Kurtosis:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-3d54si
Direct link to example
Out[1]=1

For a generalized beta distribution of the second kind, kurtosis does not depend on β:

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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-w5occx
Direct link to example
Out[2]=2

Different moments with closed forms as functions of parameters:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-js043h
Direct link to example

Moment:

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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-rx074o
Direct link to example
Out[2]=2

Closed form for symbolic order:

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In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-cq9a7c
Direct link to example
Out[3]=3

CentralMoment:

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In[4]:=4
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-pknsqa
Direct link to example
Out[4]=4

FactorialMoment:

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In[5]:=5
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-zg9ct4
Direct link to example
Out[5]=5

Cumulant:

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In[6]:=6
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-9gzmth
Direct link to example
Out[6]=6

Different moments for a generalized beta distribution of the second kind:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-ma91ii
Direct link to example

Moment:

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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-ilo25o
Direct link to example
Out[2]=2

Closed form for symbolic order:

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In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-czvwqm
Direct link to example
Out[3]=3

CentralMoment:

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In[4]:=4
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-ovua84
Direct link to example
Out[4]=4

FactorialMoment:

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In[5]:=5
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-d7e3yu
Direct link to example
Out[5]=5

Cumulant:

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In[6]:=6
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-7octsj
Direct link to example
Out[6]=6

Hazard function for a beta prime distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-k1jpte
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-tm2zss
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-vven7k
Direct link to example
Out[3]=3

Generalized beta prime distribution:

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In[4]:=4
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-bzkhpj
Direct link to example
Out[4]=4

Generalized beta distribution of the second kind:

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In[5]:=5
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-u3nopo
Direct link to example
Out[5]=5

Quantile function of a beta prime distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-nf2ugk
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-305ypi
Direct link to example
Out[2]=2

Generalized beta prime distribution:

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In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-qmfnj
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-m33ni
Direct link to example
Out[4]=4

Generalized beta distribution of the second kind:

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In[5]:=5
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-ceuozk
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-4k5ak4
Direct link to example
Out[6]=6

Consistent use of Quantity in parameters yields QuantityDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-bc5g3d
Direct link to example
Out[1]=1

Find the median loss:

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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-ep3kcu
Direct link to example
Out[2]=2

Applications  (2)Sample problems that can be solved with this function

BetaPrimeDistribution can be used to model losses:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-qytsz1
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-k3sbh9
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-pjsd2a
Direct link to example

Remove the clear outlier, Andrew, the most destructive hurricane, and attach currency units:

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In[4]:=4
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-6p37zr
Direct link to example

Fit generalized beta distribution to the data:

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In[38]:=38
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-fkjj84
Direct link to example
Out[38]=38

Compare the histogram of the data with the PDF of the estimated distribution:

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In[39]:=39
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-4tdjtb
Direct link to example
Out[39]=39

Find the probability that a loss caused by a hurricane is over 3 billion dollars:

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In[40]:=40
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-q51d0v
Direct link to example
Out[40]=40

Find the mean hurricane loss in US dollars:

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In[41]:=41
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-3oi1vl
Direct link to example
Out[41]=41

Simulate possible losses in millions of US dollars for the next 30 strong hurricanes:

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In[42]:=42
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-qp4g7z
Direct link to example
Out[42]=42

BetaPrimeDistribution can be used to model state per-capita incomes:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-1fhb3t
Direct link to example
Out[1]=1

Add currency units:

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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-m4re35
Direct link to example

Fit generalized beta distribution of the second kind to the data:

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In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-3ng636
Direct link to example
Out[3]=3

Compare the histogram of the data to the PDF of the estimated distribution:

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In[4]:=4
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-pp5e9g
Direct link to example
Out[4]=4

Find the average income per capita:

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In[5]:=5
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-w23ysw
Direct link to example
Out[5]=5

Find states with income close to the average:

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In[6]:=6
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-u9wd3b
Direct link to example
Out[6]=6

Find the median income per capita:

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In[7]:=7
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-qmsra1
Direct link to example
Out[7]=7

Find states with income close to the median:

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In[8]:=8
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-ezm0fw
Direct link to example
Out[8]=8

Find the log-likelihood value:

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In[9]:=9
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-dc8stz
Direct link to example
Out[9]=9

Properties & Relations  (16)Properties of the function, and connections to other functions

Parameter influence on the CDF of a generalized beta distribution of the second kind:

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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-ywnqzk
Direct link to example
Out[2]=2

BetaPrimeDistribution is closed under scaling by a positive factor:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-8uenm9
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-2wn5bt
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-b4juh0
Direct link to example
Out[3]=3

BetaPrimeDistribution is closed under taking inverse:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-e9dddv
Direct link to example
Out[1]=1

Relations to other distributions:

DagumDistribution is a special case of BetaPrimeDistribution:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-sjzdbv
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-idvw6v
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-2da9q0
Direct link to example
Out[3]=3

SinghMaddalaDistribution is a special case of BetaPrimeDistribution:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-ii2qh5
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-z3dkna
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-ldubh0
Direct link to example
Out[3]=3

LogLogisticDistribution is a special case of BetaPrimeDistribution:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-h0l32b
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-n117d9
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-clrien
Direct link to example
Out[3]=3

FRatioDistribution is a special case of BetaPrimeDistribution:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-nmdnvb
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-egfch9
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-431626
Direct link to example
Out[3]=3

Beta prime distribution is a special case of the type 6 PearsonDistribution:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-wtirf0
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-6w21qe
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-m78srd
Direct link to example
Out[3]=3

ParetoDistribution type II is related to BetaPrimeDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-bcb48z
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-n12jwh
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-gthe9w
Direct link to example
Out[3]=3

ParetoDistribution type IV is related to BetaPrimeDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-gg6txp
Direct link to example
Out[1]=1
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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-8hdlsa
Direct link to example
Out[2]=2
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In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-89gfwn
Direct link to example
Out[3]=3

Beta prime distribution can be obtained as a transformation of BetaDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-lw8ihh
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Out[1]=1
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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-yd91s5
Direct link to example
Out[2]=2

Generalized beta distribution of the second kind is the distribution of the ratio of two independent random variables from GammaDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-4j3n8g
Direct link to example
Out[1]=1

Generalized beta of the second kind simplifies to beta prime:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-bizm9c
Direct link to example
Out[1]=1
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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-dak2pa
Direct link to example
Out[2]=2
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In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-n6vsq2
Direct link to example
Out[3]=3

Generalized beta prime is a special case of generalized beta of the second kind:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-lg97su
Direct link to example
Out[1]=1
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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-2k9qh2
Direct link to example
Out[2]=2
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In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-pvbxjd
Direct link to example
Out[3]=3

Generalized beta prime simplifies to beta prime distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-51ze39
Direct link to example
Out[1]=1
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In[2]:=2
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-kh0xhz
Direct link to example
Out[2]=2
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In[3]:=3
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-lkebd0
Direct link to example
Out[3]=3

Neat Examples  (1)Surprising or curious use cases

PDFs for different q values with CDF contours:

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In[1]:=1
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-io0104
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e5c3ivjl42rg6-ceikus
Direct link to example
Out[4]=4
Wolfram Research (2010), BetaPrimeDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaPrimeDistribution.html (updated 2016).
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Wolfram Research (2010), BetaPrimeDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaPrimeDistribution.html (updated 2016).

Text

Wolfram Research (2010), BetaPrimeDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaPrimeDistribution.html (updated 2016).

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Wolfram Research (2010), BetaPrimeDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaPrimeDistribution.html (updated 2016).

CMS

Wolfram Language. 2010. "BetaPrimeDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/BetaPrimeDistribution.html.

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Wolfram Language. 2010. "BetaPrimeDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/BetaPrimeDistribution.html.

APA

Wolfram Language. (2010). BetaPrimeDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BetaPrimeDistribution.html

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Wolfram Language. (2010). BetaPrimeDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BetaPrimeDistribution.html

BibTeX

@misc{reference.wolfram_2025_betaprimedistribution, author="Wolfram Research", title="{BetaPrimeDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/BetaPrimeDistribution.html}", note=[Accessed: 26-March-2025 ]}

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@misc{reference.wolfram_2025_betaprimedistribution, author="Wolfram Research", title="{BetaPrimeDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/BetaPrimeDistribution.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_betaprimedistribution, organization={Wolfram Research}, title={BetaPrimeDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/BetaPrimeDistribution.html}, note=[Accessed: 26-March-2025 ]}

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@online{reference.wolfram_2025_betaprimedistribution, organization={Wolfram Research}, title={BetaPrimeDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/BetaPrimeDistribution.html}, note=[Accessed: 26-March-2025 ]}