# Boost Math - Statistical Distributions

## Statistics

The Statistical Distributions section of the Boost Math library cover a broad range of areas

### Arcsine Distribution

    # Arcsine distribution with default parameters
    dist <- arcsine_distribution()
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.5
    logcdf(dist, 0.5)
    #> [1] -0.6931472
    pdf(dist, 0.5)
    #> [1] 0.6366198
    logpdf(dist, 0.5)
    #> [1] -0.4515827
    hazard(dist, 0.5)
    #> [1] 1.27324
    chf(dist, 0.5)
    #> [1] 0.6931472
    mean(dist)
    #> [1] 0.5
    median(dist)
    #> [1] 0.5
    range(dist)
    #> [1] 0 1
    quantile(dist, 0.2)
    #> [1] 0.0954915
    standard_deviation(dist)
    #> [1] 0.3535534
    support(dist)
    #> [1] 0 1
    variance(dist)
    #> [1] 0.125
    skewness(dist)
    #> [1] 0
    kurtosis(dist)
    #> [1] 1.5
    kurtosis_excess(dist)
    #> [1] -1.5

    # Convenience functions
    arcsine_pdf(0.5)
    #> [1] 0.6366198
    arcsine_lpdf(0.5)
    #> [1] -0.4515827
    arcsine_cdf(0.5)
    #> [1] 0.5
    arcsine_lcdf(0.5)
    #> [1] -0.6931472
    arcsine_quantile(0.5)
    #> [1] 0.5

### Bernoulli Distribution

    # Bernoulli distribution with p_success = 0.5
    dist <- bernoulli_distribution(0.5)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.8183099
    logcdf(dist, 0.5)
    #> [1] -0.2005142
    pdf(dist, 0.5)
    #> [1] 0.6366198
    logpdf(dist, 0.5)
    #> [1] -0.4515827
    hazard(dist, 0.5)
    #> [1] 3.503877
    chf(dist, 0.5)
    #> [1] 1.705453
    mean(dist)
    #> [1] 0.5
    median(dist)
    #> [1] 0
    mode(dist)
    #> [1] 1
    range(dist)
    #> [1] 0 1
    quantile(dist, 0.2)
    #> [1] 0
    standard_deviation(dist)
    #> [1] 0.5
    support(dist)
    #> [1] 0 1
    variance(dist)
    #> [1] 0.25
    skewness(dist)
    #> [1] 0
    kurtosis(dist)
    #> [1] 1
    kurtosis_excess(dist)
    #> [1] -2

    # Convenience functions
    bernoulli_pdf(1, 0.5)
    #> [1] 0.5
    bernoulli_lpdf(1, 0.5)
    #> [1] -0.6931472
    bernoulli_cdf(1, 0.5)
    #> [1] 1
    bernoulli_lcdf(1, 0.5)
    #> [1] 0
    bernoulli_quantile(0.5, 0.5)
    #> [1] 0

### Beta Distribution

    # Beta distribution with shape parameters alpha = 2, beta = 5
    dist <- beta_distribution(2, 5)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.890625
    logcdf(dist, 0.5)
    #> [1] -0.1158318
    pdf(dist, 0.5)
    #> [1] 0.9375
    logpdf(dist, 0.5)
    #> [1] -0.06453852
    hazard(dist, 0.5)
    #> [1] 8.571429
    chf(dist, 0.5)
    #> [1] 2.212973
    mean(dist)
    #> [1] 0.2857143
    median(dist)
    #> [1] 0.26445
    mode(dist)
    #> [1] 0.2
    range(dist)
    #> [1] 0 1
    quantile(dist, 0.2)
    #> [1] 0.1398807
    standard_deviation(dist)
    #> [1] 0.1597191
    support(dist)
    #> [1] 0 1
    variance(dist)
    #> [1] 0.0255102
    skewness(dist)
    #> [1] 0.5962848
    kurtosis(dist)
    #> [1] 2.88
    kurtosis_excess(dist)
    #> [1] -0.12

    # Convenience functions
    beta_pdf(0.5, 2, 5)
    #> [1] 0.9375
    beta_lpdf(0.5, 2, 5)
    #> [1] -0.06453852
    beta_cdf(0.5, 2, 5)
    #> [1] 0.890625
    beta_lcdf(0.5, 2, 5)
    #> [1] -0.1158318
    beta_quantile(0.5, 2, 5)
    #> [1] 0.26445

### Binomial Distribution

    # Binomial dist ribution with n = 10, prob = 0.5
    dist <- binomial_distribution(10, 0.5)
    # Apply generic functions
    cdf(dist, 2)
    #> [1] 0.0546875
    logcdf(dist, 2)
    #> [1] -2.90612
    pdf(dist, 2)
    #> [1] 0.04394531
    logpdf(dist, 2)
    #> [1] -3.124809
    hazard(dist, 2)
    #> [1] 0.0464876
    chf(dist, 2)
    #> [1] 0.05623972
    mean(dist)
    #> [1] 5
    median(dist)
    #> [1] 5
    mode(dist)
    #> [1] 5
    range(dist)
    #> [1]  0 10
    quantile(dist, 0.2)
    #> [1] 3
    standard_deviation(dist)
    #> [1] 1.581139
    support(dist)
    #> [1]  0 10
    variance(dist)
    #> [1] 2.5
    skewness(dist)
    #> [1] 0
    kurtosis(dist)
    #> [1] 2.8
    kurtosis_excess(dist)
    #> [1] -0.2

    # Convenience functions
    binomial_pdf(3, 10, 0.5)
    #> [1] 0.1171875
    binomial_lpdf(3, 10, 0.5)
    #> [1] -2.14398
    binomial_cdf(3, 10, 0.5)
    #> [1] 0.171875
    binomial_lcdf(3, 10, 0.5)
    #> [1] -1.760988
    binomial_quantile(0.5, 10, 0.5)
    #> [1] 5

### Cauchy Distribution

    # Cauchy distribution with location = 0, scale = 1
    dist <- cauchy_distribution(0, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.6475836
    logcdf(dist, 0.5)
    #> [1] -0.4345074
    pdf(dist, 0.5)
    #> [1] 0.2546479
    logpdf(dist, 0.5)
    #> [1] -1.367873
    hazard(dist, 0.5)
    #> [1] 0.7225768
    chf(dist, 0.5)
    #> [1] 1.042942
    median(dist)
    #> [1] 0
    mode(dist)
    #> [1] 0
    range(dist)
    #> [1] -Inf  Inf
    quantile(dist, 0.2)
    #> [1] -1.376382
    support(dist)
    #> [1] -Inf  Inf

    # Convenience functions
    cauchy_pdf(0)
    #> [1] 0.3183099
    cauchy_lpdf(0)
    #> [1] -1.14473
    cauchy_cdf(0)
    #> [1] 0.5
    cauchy_lcdf(0)
    #> [1] -0.6931472
    cauchy_quantile(0.5)
    #> [1] 0

### Chi-Squared Distribution

    # Chi-Squared distribution with 3 degrees of freedom
    dist <- chi_squared_distribution(3)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.08110859
    logcdf(dist, 0.5)
    #> [1] -2.511966
    pdf(dist, 0.5)
    #> [1] 0.2196956
    logpdf(dist, 0.5)
    #> [1] -1.515512
    hazard(dist, 0.5)
    #> [1] 0.2390877
    chf(dist, 0.5)
    #> [1] 0.08458732
    mean(dist)
    #> [1] 3
    median(dist)
    #> [1] 2.365974
    mode(dist)
    #> [1] 1
    range(dist)
    #> [1]   0 Inf
    quantile(dist, 0.2)
    #> [1] 1.005174
    standard_deviation(dist)
    #> [1] 2.44949
    support(dist)
    #> [1]  0.000000e+00 1.797693e+308
    variance(dist)
    #> [1] 6
    skewness(dist)
    #> [1] 1.632993
    kurtosis(dist)
    #> [1] 7
    kurtosis_excess(dist)
    #> [1] 4

    # Convenience functions
    chi_squared_pdf(2, 3)
    #> [1] 0.2075537
    chi_squared_lpdf(2, 3)
    #> [1] -1.572365
    chi_squared_cdf(2, 3)
    #> [1] 0.4275933
    chi_squared_lcdf(2, 3)
    #> [1] -0.8495828
    chi_squared_quantile(0.5, 3)
    #> [1] 2.365974

### Exponential Distribution

    # Exponential distribution with rate parameter lambda = 2
    dist <- exponential_distribution(2)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.6321206
    logcdf(dist, 0.5)
    #> [1] -0.4586751
    pdf(dist, 0.5)
    #> [1] 0.7357589
    logpdf(dist, 0.5)
    #> [1] -0.3068528
    hazard(dist, 0.5)
    #> [1] 2
    chf(dist, 0.5)
    #> [1] 1
    mean(dist)
    #> [1] 0.5
    median(dist)
    #> [1] 0.3465736
    mode(dist)
    #> [1] 0
    range(dist)
    #> [1]   0 Inf
    quantile(dist, 0.2)
    #> [1] 0.1115718
    standard_deviation(dist)
    #> [1] 0.5
    support(dist)
    #> [1] 2.225074e-308 1.797693e+308
    variance(dist)
    #> [1] 0.25
    skewness(dist)
    #> [1] 2
    kurtosis(dist)
    #> [1] 9
    kurtosis_excess(dist)
    #> [1] 6

    # Convenience functions
    exponential_pdf(1, 2)
    #> [1] 0.2706706
    exponential_lpdf(1, 2)
    #> [1] -1.306853
    exponential_cdf(1, 2)
    #> [1] 0.8646647
    exponential_lcdf(1, 2)
    #> [1] -0.1454135
    exponential_quantile(0.5, 2)
    #> [1] 0.3465736

### Extreme Value Distribution

    # Extreme Value distribution with location = 0, scale = 1
    dist <- extreme_value_distribution(0, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.5452392
    logcdf(dist, 0.5)
    #> [1] -0.6065307
    pdf(dist, 0.5)
    #> [1] 0.3307043
    logpdf(dist, 0.5)
    #> [1] -1.106531
    hazard(dist, 0.5)
    #> [1] 0.727205
    chf(dist, 0.5)
    #> [1] 0.7879837
    mean(dist)
    #> [1] 0.5772157
    median(dist)
    #> [1] 0.3665129
    mode(dist)
    #> [1] 0
    range(dist)
    #> [1] -Inf  Inf
    quantile(dist, 0.2)
    #> [1] -0.475885
    standard_deviation(dist)
    #> [1] 1.28255
    support(dist)
    #> [1] -1.797693e+308  1.797693e+308
    variance(dist)
    #> [1] 1.644934
    skewness(dist)
    #> [1] 1.139547
    kurtosis(dist)
    #> [1] 5.4
    kurtosis_excess(dist)
    #> [1] 2.4

    # Convenience functions
    extreme_value_pdf(0)
    #> [1] 0.3678794
    extreme_value_lpdf(0)
    #> [1] -1
    extreme_value_cdf(0)
    #> [1] 0.3678794
    extreme_value_lcdf(0)
    #> [1] -1
    extreme_value_quantile(0.5)
    #> [1] 0.3665129

### F Distribution

    # Fisher F distribution with df1 = 5, df2 = 10
    dist <- fisher_f_distribution(5, 10)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.2299751
    logcdf(dist, 0.5)
    #> [1] -1.469784
    pdf(dist, 0.5)
    #> [1] 0.687607
    logpdf(dist, 0.5)
    #> [1] -0.3745378
    hazard(dist, 0.5)
    #> [1] 0.8929673
    chf(dist, 0.5)
    #> [1] 0.2613325
    mean(dist)
    #> [1] 1.25
    median(dist)
    #> [1] 0.9319332
    mode(dist)
    #> [1] 0.5
    range(dist)
    #> [1]  0.000000e+00 1.797693e+308
    quantile(dist, 0.2)
    #> [1] 0.4563364
    standard_deviation(dist)
    #> [1] 1.163687
    support(dist)
    #> [1]  0.000000e+00 1.797693e+308
    variance(dist)
    #> [1] 1.354167
    skewness(dist)
    #> [1] 3.86702
    kurtosis(dist)
    #> [1] 53.86154
    kurtosis_excess(dist)
    #> [1] 50.86154

    # Convenience functions
    fisher_f_pdf(1, 5, 10)
    #> [1] 0.4954798
    fisher_f_lpdf(1, 5, 10)
    #> [1] -0.7022287
    fisher_f_cdf(1, 5, 10)
    #> [1] 0.5348806
    fisher_f_lcdf(1, 5, 10)
    #> [1] -0.6257118
    fisher_f_quantile(0.5, 5, 10)
    #> [1] 0.9319332

### Gamma Distribution

    # Gamma distribution with shape = 3, scale = 4
    dist <- gamma_distribution(3, 4)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.0002964775
    logcdf(dist, 0.5)
    #> [1] -8.123539
    pdf(dist, 0.5)
    #> [1] 0.001723627
    logpdf(dist, 0.5)
    #> [1] -6.363325
    hazard(dist, 0.5)
    #> [1] 0.001724138
    chf(dist, 0.5)
    #> [1] 0.0002965215
    mean(dist)
    #> [1] 12
    median(dist)
    #> [1] 10.69624
    mode(dist)
    #> [1] 8
    range(dist)
    #> [1]  0.000000e+00 1.797693e+308
    quantile(dist, 0.2)
    #> [1] 6.140177
    standard_deviation(dist)
    #> [1] 6.928203
    support(dist)
    #> [1] 2.225074e-308 1.797693e+308
    variance(dist)
    #> [1] 48
    skewness(dist)
    #> [1] 1.154701
    kurtosis(dist)
    #> [1] 5
    kurtosis_excess(dist)
    #> [1] 2

    # Convenience functions
    gamma_pdf(2, 3, 4)
    #> [1] 0.01895408
    gamma_lpdf(2, 3, 4)
    #> [1] -3.965736
    gamma_cdf(2, 3, 4)
    #> [1] 0.01438768
    gamma_lcdf(2, 3, 4)
    #> [1] -4.241383
    gamma_quantile(0.5, 3, 4)
    #> [1] 10.69624

### Geometric Distribution

    # Geometric distribution with probability of success prob = 0.5
    dist <- geometric_distribution(0.5)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.6464466
    logcdf(dist, 0.5)
    #> [1] -0.4362647
    pdf(dist, 0.5)
    #> [1] 0.3535534
    logpdf(dist, 0.5)
    #> [1] -1.039721
    hazard(dist, 0.5)
    #> [1] 1
    chf(dist, 0.5)
    #> [1] 1.039721
    mean(dist)
    #> [1] 1
    median(dist)
    #> [1] 0
    mode(dist)
    #> [1] 0
    range(dist)
    #> [1]  0.000000e+00 1.797693e+308
    quantile(dist, 0.2)
    #> [1] 0
    standard_deviation(dist)
    #> [1] 1.414214
    support(dist)
    #> [1]  0.000000e+00 1.797693e+308
    variance(dist)
    #> [1] 2
    skewness(dist)
    #> [1] 2.12132
    kurtosis(dist)
    #> [1] 9.5
    kurtosis_excess(dist)
    #> [1] 6.5

    # Convenience functions
    geometric_pdf(3, 0.5)
    #> [1] 0.0625
    geometric_lpdf(3, 0.5)
    #> [1] -2.772589
    geometric_cdf(3, 0.5)
    #> [1] 0.9375
    geometric_lcdf(3, 0.5)
    #> [1] -0.06453852
    geometric_quantile(0.5, 0.5)
    #> [1] 0

### Holtsmark Distribution

    # Holtsmark distribution with location 0 and scale 1
    dist <- holtsmark_distribution(0, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.6394042
    logcdf(dist, 0.5)
    #> [1] -0.4472184
    pdf(dist, 0.5)
    #> [1] 0.2622968
    logpdf(dist, 0.5)
    #> [1] -1.338278
    hazard(dist, 0.5)
    #> [1] 0.7273985
    chf(dist, 0.5)
    #> [1] 1.019998
    mean(dist)
    #> [1] 0
    median(dist)
    #> [1] 0
    mode(dist)
    #> [1] 0
    range(dist)
    #> [1] -Inf  Inf
    quantile(dist, 0.2)
    #> [1] -1.234616
    standard_deviation(dist)
    #> [1] Inf
    support(dist)
    #> [1] -Inf  Inf
    variance(dist)
    #> [1] Inf

    # Convenience functions
    holtsmark_pdf(3)
    #> [1] 0.03150942
    holtsmark_lpdf(3)
    #> [1] -3.457469
    holtsmark_cdf(3)
    #> [1] 0.9484022
    holtsmark_lcdf(3)
    #> [1] -0.05297661
    holtsmark_quantile(0.5)
    #> [1] 0

### Hyperexponential Distribution

    # Hyperexponential distribution with probabilities = c(0.5, 0.5) and rates = c(1, 2)
    dist <- hyperexponential_distribution(c(0.5, 0.5), c(1, 2))
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.5127949
    logcdf(dist, 0.5)
    #> [1] -0.6678792
    pdf(dist, 0.5)
    #> [1] 0.6711448
    logpdf(dist, 0.5)
    #> [1] -0.3987704
    hazard(dist, 0.5)
    #> [1] 1.377541
    chf(dist, 0.5)
    #> [1] 0.7190702
    mean(dist)
    #> [1] 0.75
    median(dist)
    #> [1] 0.4812118
    mode(dist)
    #> [1] 0
    range(dist)
    #> [1]   0 Inf
    quantile(dist, 0.2)
    #> [1] 0.1506519
    standard_deviation(dist)
    #> [1] 0.8291562
    support(dist)
    #> [1] 2.225074e-308 1.797693e+308
    variance(dist)
    #> [1] 0.6875
    skewness(dist)
    #> [1] 2.466911
    kurtosis(dist)
    #> [1] 12.47107
    kurtosis_excess(dist)
    #> [1] 9.471074

    # Convenience functions
    hyperexponential_pdf(2, c(0.5, 0.5), c(1, 2))
    #> [1] 0.08598328
    hyperexponential_lpdf(2, c(0.5, 0.5), c(1, 2))
    #> [1] -2.453602
    hyperexponential_cdf(2, c(0.5, 0.5), c(1, 2))
    #> [1] 0.9231745
    hyperexponential_lcdf(2, c(0.5, 0.5), c(1, 2))
    #> [1] -0.07993696
    hyperexponential_quantile(0.5, c(0.5, 0.5), c(1, 2))
    #> [1] 0.4812118

### Hypergeometric Distribution

    # Hypergeometric distribution with r = 5, n = 10, N = 20
    dist <- hypergeometric_distribution(5, 10, 20)
    # Apply generic functions
    cdf(dist, 4)
    #> [1] 0.9837461
    logcdf(dist, 4)
    #> [1] -0.01638741
    pdf(dist, 4)
    #> [1] 0.1354489
    logpdf(dist, 4)
    #> [1] -1.999161
    hazard(dist, 4)
    #> [1] 8.333333
    chf(dist, 4)
    #> [1] 4.119424
    mean(dist)
    #> [1] 2.5
    median(dist)
    #> [1] 3
    mode(dist)
    #> [1] 3
    range(dist)
    #> [1] 0 5
    quantile(dist, 0.2)
    #> [1] 1
    standard_deviation(dist)
    #> [1] 0.9933993
    support(dist)
    #> [1] 0 5
    variance(dist)
    #> [1] 0.9868421
    skewness(dist)
    #> [1] 0
    kurtosis(dist)
    #> [1] 5.756863
    kurtosis_excess(dist)
    #> [1] 2.756863

    # Convenience functions
    hypergeometric_pdf(3, 5, 10, 20)
    #> [1] 0.3482972
    hypergeometric_lpdf(3, 5, 10, 20)
    #> [1] -1.054699
    hypergeometric_cdf(3, 5, 10, 20)
    #> [1] 0.8482972
    hypergeometric_lcdf(3, 5, 10, 20)
    #> [1] -0.1645242
    hypergeometric_quantile(0.5, 5, 10, 20)
    #> [1] 3

### Inverse Chi-Squared Distribution

    # Inverse Chi-Squared distribution with 10 degrees of freedom, scale = 1
    dist <- inverse_chi_squared_distribution(10, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.02925269
    logcdf(dist, 0.5)
    #> [1] -3.531784
    pdf(dist, 0.5)
    #> [1] 0.3783327
    logpdf(dist, 0.5)
    #> [1] -0.9719812
    hazard(dist, 0.5)
    #> [1] 0.3897335
    chf(dist, 0.5)
    #> [1] 0.02968908
    mean(dist)
    #> [1] 1.25
    median(dist)
    #> [1] 1.070455
    mode(dist)
    #> [1] 0.8333333
    range(dist)
    #> [1]  0.000000e+00 1.797693e+308
    quantile(dist, 0.2)
    #> [1] 0.7439393
    standard_deviation(dist)
    #> [1] 0.7216878
    support(dist)
    #> [1]  0.000000e+00 1.797693e+308
    variance(dist)
    #> [1] 0.5208333
    skewness(dist)
    #> [1] 3.464102
    kurtosis(dist)
    #> [1] 45
    kurtosis_excess(dist)
    #> [1] 42

    # Convenience functions
    inverse_chi_squared_pdf(2, 10, 1)
    #> [1] 0.1670024
    inverse_chi_squared_lpdf(2, 10, 1)
    #> [1] -1.789747
    inverse_chi_squared_cdf(2, 10, 1)
    #> [1] 0.891178
    inverse_chi_squared_lcdf(2, 10, 1)
    #> [1] -0.1152111
    inverse_chi_squared_quantile(0.5, 10, 1)
    #> [1] 1.070455

### Inverse Gamma Distribution

    # Inverse Gamma distribution with shape = 5, scale = 4
    dist <- inverse_gamma_distribution(5, 4)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.0996324
    logcdf(dist, 0.5)
    #> [1] -2.306268
    pdf(dist, 0.5)
    #> [1] 0.9160366
    logpdf(dist, 0.5)
    #> [1] -0.08769894
    hazard(dist, 0.5)
    #> [1] 1.017403
    chf(dist, 0.5)
    #> [1] 0.1049522
    mean(dist)
    #> [1] 1
    median(dist)
    #> [1] 0.8563644
    mode(dist)
    #> [1] 0.6666667
    range(dist)
    #> [1]  0.000000e+00 1.797693e+308
    quantile(dist, 0.2)
    #> [1] 0.5951514
    standard_deviation(dist)
    #> [1] 0.5773503
    support(dist)
    #> [1]  0.000000e+00 1.797693e+308
    variance(dist)
    #> [1] 0.3333333
    skewness(dist)
    #> [1] 3.464102
    kurtosis(dist)
    #> [1] 45
    kurtosis_excess(dist)
    #> [1] 42

    # Convenience functions
    inverse_gamma_pdf(2, 5, 4)
    #> [1] 0.09022352
    inverse_gamma_lpdf(2, 5, 4)
    #> [1] -2.405465
    inverse_gamma_cdf(2, 5, 4)
    #> [1] 0.947347
    inverse_gamma_lcdf(2, 5, 4)
    #> [1] -0.05408985
    inverse_gamma_quantile(0.5, 5, 4)
    #> [1] 0.8563644

### Inverse Gaussian Distribution

    # Inverse Gaussian distribution with mu = 3, lambda = 4
    dist <- inverse_gaussian_distribution(3, 4)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.01617264
    logcdf(dist, 0.5)
    #> [1] -4.124435
    pdf(dist, 0.5)
    #> [1] 0.1403174
    logpdf(dist, 0.5)
    #> [1] -1.963848
    hazard(dist, 0.5)
    #> [1] 0.142624
    chf(dist, 0.5)
    #> [1] 0.01630484
    mean(dist)
    #> [1] 3
    median(dist)
    #> [1] 2.202698
    mode(dist)
    #> [1] 1.140598
    range(dist)
    #> [1]  0.000000e+00 1.797693e+308
    quantile(dist, 0.2)
    #> [1] 1.161488
    standard_deviation(dist)
    #> [1] 2.598076
    support(dist)
    #> [1]  0.000000e+00 1.797693e+308
    variance(dist)
    #> [1] 6.75
    skewness(dist)
    #> [1] 2.598076
    kurtosis(dist)
    #> [1] 8.25
    kurtosis_excess(dist)
    #> [1] 11.25

    # Convenience functions
    inverse_gaussian_pdf(2, 3, 4)
    #> [1] 0.2524295
    inverse_gaussian_lpdf(2, 3, 4)
    #> [1] -1.376623
    inverse_gaussian_cdf(2, 3, 4)
    #> [1] 0.4512408
    inverse_gaussian_lcdf(2, 3, 4)
    #> [1] -0.7957542
    inverse_gaussian_quantile(0.5, 3, 4)
    #> [1] 2.202698

### Kolmogorov-Smirnov Distribution

    # Kolmogorov-Smirnov distribution with sample size n = 10
    dist <- kolmogorov_smirnov_distribution(10)
    # Apply generic functions
    cdf(dist, 2)
    #> [1] 1
    logcdf(dist, 2)
    #> [1] 0
    pdf(dist, 2)
    #> [1] 2.887762e-33
    logpdf(dist, 2)
    #> [1] -74.92483
    hazard(dist, 2)
    #> [1] 80
    chf(dist, 2)
    #> [1] 79.30685
    mean(dist)
    #> [1] 0.2747169
    median(dist)
    #> [1] 0.2617017
    mode(dist)
    #> [1] 0.2325754
    range(dist)
    #> [1]  0.000000e+00 1.797693e+308
    quantile(dist, 0.2)
    #> [1] 0.2039076
    standard_deviation(dist)
    #> [1] 0.08232448
    support(dist)
    #> [1]  0.000000e+00 1.797693e+308
    variance(dist)
    #> [1] 0.00677732
    skewness(dist)
    #> [1] 0.8604261
    kurtosis(dist)
    #> [1] 3.881619
    kurtosis_excess(dist)
    #> [1] 0.881619

    # Convenience functions
    kolmogorov_smirnov_pdf(0.5, 10)
    #> [1] 0.2695176
    kolmogorov_smirnov_lpdf(0.5, 10)
    #> [1] -1.311122
    kolmogorov_smirnov_cdf(0.5, 10)
    #> [1] 0.9865241
    kolmogorov_smirnov_lcdf(0.5, 10)
    #> [1] -0.01356751
    kolmogorov_smirnov_quantile(0.5, 10)
    #> [1] 0.2617017

### Landau Distribution

    # Landau distribution with location 0 and scale 1
    dist <- landau_distribution(0, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.4842393
    logcdf(dist, 0.5)
    #> [1] -0.7251762
    pdf(dist, 0.5)
    #> [1] 0.2123185
    logpdf(dist, 0.5)
    #> [1] -1.549668
    hazard(dist, 0.5)
    #> [1] 0.4116608
    chf(dist, 0.5)
    #> [1] 0.6621123
    median(dist)
    #> [1] 0.5756301
    mode(dist)
    #> [1] -0.4293145
    range(dist)
    #> [1] -Inf  Inf
    quantile(dist, 0.2)
    #> [1] -0.5948319
    support(dist)
    #> [1] -Inf  Inf

    # Convenience functions
    landau_pdf(3)
    #> [1] 0.05863949
    landau_lpdf(3)
    #> [1] -2.836347
    landau_cdf(3)
    #> [1] 0.7792967
    landau_lcdf(3)
    #> [1] -0.2493635
    landau_quantile(0.5)
    #> [1] 0.5756301

### Laplace Distribution

    # Laplace distribution with location = 0, scale = 1
    dist <- laplace_distribution(0, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.6967347
    logcdf(dist, 0.5)
    #> [1] -0.3613506
    pdf(dist, 0.5)
    #> [1] 0.3032653
    logpdf(dist, 0.5)
    #> [1] -1.193147
    hazard(dist, 0.5)
    #> [1] 1
    chf(dist, 0.5)
    #> [1] 1.193147
    mean(dist)
    #> [1] 0
    median(dist)
    #> [1] 0
    mode(dist)
    #> [1] 0
    range(dist)
    #> [1] -Inf  Inf
    quantile(dist, 0.2)
    #> [1] -0.9162907
    standard_deviation(dist)
    #> [1] 1.414214
    support(dist)
    #> [1] -Inf  Inf
    variance(dist)
    #> [1] 2
    skewness(dist)
    #> [1] 0
    kurtosis(dist)
    #> [1] 6
    kurtosis_excess(dist)
    #> [1] 3

    # Convenience functions
    laplace_pdf(0)
    #> [1] 0.5
    laplace_lpdf(0)
    #> [1] -0.6931472
    laplace_cdf(0)
    #> [1] 0.5
    laplace_lcdf(0)
    #> [1] -0.6931472
    laplace_quantile(0.5)
    #> [1] 0

### Logistic Distribution

    # Logistic distribution with location = 0, scale = 1
    dist <- logistic_distribution(0, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.6224593
    logcdf(dist, 0.5)
    #> [1] -0.474077
    pdf(dist, 0.5)
    #> [1] 0.2350037
    logpdf(dist, 0.5)
    #> [1] -1.448154
    hazard(dist, 0.5)
    #> [1] 0.6224593
    chf(dist, 0.5)
    #> [1] 0.974077
    mean(dist)
    #> [1] 0
    median(dist)
    #> [1] 0
    mode(dist)
    #> [1] 0
    range(dist)
    #> [1] -Inf  Inf
    quantile(dist, 0.2)
    #> [1] -1.386294
    standard_deviation(dist)
    #> [1] 1.813799
    support(dist)
    #> [1] -1.797693e+308  1.797693e+308
    variance(dist)
    #> [1] 3.289868
    skewness(dist)
    #> [1] 0
    kurtosis(dist)
    #> [1] 4.2
    kurtosis_excess(dist)
    #> [1] 1.2

    # Convenience functions
    logistic_pdf(0)
    #> [1] 0.25
    logistic_lpdf(0)
    #> [1] -1.386294
    logistic_cdf(0)
    #> [1] 0.5
    logistic_lcdf(0)
    #> [1] -0.6931472
    logistic_quantile(0.5)
    #> [1] 0

### Log Normal Distribution

    # Log Normal distribution with location = 0, scale = 1
    dist <- lognormal_distribution(0, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.2441086
    logcdf(dist, 0.5)
    #> [1] -1.410142
    pdf(dist, 0.5)
    #> [1] 0.6274961
    logpdf(dist, 0.5)
    #> [1] -0.4660179
    hazard(dist, 0.5)
    #> [1] 0.8301405
    chf(dist, 0.5)
    #> [1] 0.2798576
    mean(dist)
    #> [1] 1.648721
    median(dist)
    #> [1] 1
    mode(dist)
    #> [1] 0.3678794
    range(dist)
    #> [1]  0.000000e+00 1.797693e+308
    quantile(dist, 0.2)
    #> [1] 0.4310112
    standard_deviation(dist)
    #> [1] 2.161197
    support(dist)
    #> [1]  0.000000e+00 1.797693e+308
    variance(dist)
    #> [1] 4.670774
    skewness(dist)
    #> [1] 6.184877
    kurtosis(dist)
    #> [1] 113.9364
    kurtosis_excess(dist)
    #> [1] 110.9364

    # Convenience functions
    lognormal_pdf(0)
    #> [1] 0
    lognormal_lpdf(0)
    #> [1] -Inf
    lognormal_cdf(0)
    #> [1] 0
    lognormal_lcdf(0)
    #> [1] -Inf
    lognormal_quantile(0.5)
    #> [1] 1

### Map-Airy Distribution

    # Map-Airy distribution with location 0 and scale 1
    dist <- mapairy_distribution(0, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.7527466
    logcdf(dist, 0.5)
    #> [1] -0.2840266
    pdf(dist, 0.5)
    #> [1] 0.1477024
    logpdf(dist, 0.5)
    #> [1] -1.912556
    hazard(dist, 0.5)
    #> [1] 0.5973726
    chf(dist, 0.5)
    #> [1] 1.397342
    mean(dist)
    #> [1] 0
    median(dist)
    #> [1] -0.7167107
    mode(dist)
    #> [1] -1.161587
    range(dist)
    #> [1] -Inf  Inf
    quantile(dist, 0.2)
    #> [1] -1.834427
    standard_deviation(dist)
    #> [1] Inf
    support(dist)
    #> [1] -Inf  Inf
    variance(dist)
    #> [1] Inf

    # Convenience functions
    mapairy_pdf(3)
    #> [1] 0.02799732
    mapairy_lpdf(3)
    #> [1] -3.575647
    mapairy_cdf(3)
    #> [1] 0.9316961
    mapairy_lcdf(3)
    #> [1] -0.07074859
    mapairy_quantile(0.5)
    #> [1] -0.7167107

### Negative Binomial Distribution

    # Negative Binomial distribution with successes = 5, success_fraction = 0.5
    dist <- negative_binomial_distribution(5, 0.5)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.06449911
    logcdf(dist, 0.5)
    #> [1] -2.741104
    pdf(dist, 0.5)
    #> [1] 0.05437955
    logpdf(dist, 0.5)
    #> [1] -2.911767
    hazard(dist, 0.5)
    #> [1] 0.05812881
    chf(dist, 0.5)
    #> [1] 0.06667318
    mean(dist)
    #> [1] 5
    median(dist)
    #> [1] 4
    mode(dist)
    #> [1] 4
    range(dist)
    #> [1]  0.000000e+00 1.797693e+308
    quantile(dist, 0.2)
    #> [1] 1
    standard_deviation(dist)
    #> [1] 3.162278
    support(dist)
    #> [1]  0.000000e+00 1.797693e+308
    variance(dist)
    #> [1] 10
    skewness(dist)
    #> [1] 0.9486833
    kurtosis(dist)
    #> [1] 4.3
    kurtosis_excess(dist)
    #> [1] 1.3

    # Convenience functions
    negative_binomial_pdf(3, 5, 0.5)
    #> [1] 0.1367187
    negative_binomial_lpdf(3, 5, 0.5)
    #> [1] -1.989829
    negative_binomial_cdf(3, 5, 0.5)
    #> [1] 0.3632812
    negative_binomial_lcdf(3, 5, 0.5)
    #> [1] -1.012578
    negative_binomial_quantile(0.5, 5, 0.5)
    #> [1] 4

### Noncentral Beta Distribution

    # Noncentral Beta distribution with shape parameters alpha = 2, beta = 3
    # and noncentrality parameter lambda = 1
    dist <- non_central_beta_distribution(2, 3, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.5977904
    logcdf(dist, 0.5)
    #> [1] -0.514515
    pdf(dist, 0.5)
    #> [1] 1.643543
    logpdf(dist, 0.5)
    #> [1] 0.4968546
    hazard(dist, 0.5)
    #> [1] 4.086286
    chf(dist, 0.5)
    #> [1] 0.910782
    mean(dist)
    #> [1] 0.44664
    median(dist)
    #> [1] 0.4416064
    mode(dist)
    #> [1] 0.4262677
    range(dist)
    #> [1] 0 1
    quantile(dist, 0.2)
    #> [1] 0.2549084
    standard_deviation(dist)
    #> [1] 0.2040433
    support(dist)
    #> [1] 0 1
    variance(dist)
    #> [1] 0.04163366

    # Convenience functions
    non_central_beta_pdf(0.5, 2, 3, 1)
    #> [1] 1.643543
    non_central_beta_lpdf(0.5, 2, 3, 1)
    #> [1] 0.4968546
    non_central_beta_cdf(0.5, 2, 3, 1)
    #> [1] 0.5977904
    non_central_beta_lcdf(0.5, 2, 3, 1)
    #> [1] -0.514515
    non_central_beta_quantile(0.5, 2, 3, 1)
    #> [1] 0.4416064

### Noncentral Chi-Squared Distribution

    # Noncentral Chi-Squared distribution with 3 degrees of freedom and noncentrality
    # parameter 1
    dist <- non_central_chi_squared_distribution(3, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.05162593
    logcdf(dist, 0.5)
    #> [1] -2.963731
    pdf(dist, 0.5)
    #> [1] 0.1446374
    logpdf(dist, 0.5)
    #> [1] -1.933525
    hazard(dist, 0.5)
    #> [1] 0.1525109
    chf(dist, 0.5)
    #> [1] 0.05300627
    mean(dist)
    #> [1] 4
    median(dist)
    #> [1] 3.213009
    mode(dist)
    #> [1] 1.439229
    range(dist)
    #> [1]  0.000000e+00 1.797693e+308
    quantile(dist, 0.2)
    #> [1] 1.384937
    standard_deviation(dist)
    #> [1] 3.162278
    support(dist)
    #> [1]  0.000000e+00 1.797693e+308
    variance(dist)
    #> [1] 10
    skewness(dist)
    #> [1] 1.517893
    kurtosis(dist)
    #> [1] 6.36
    kurtosis_excess(dist)
    #> [1] 3.36

    # Convenience functions
    non_central_chi_squared_pdf(2, 3, 1)
    #> [1] 0.172252
    non_central_chi_squared_lpdf(2, 3, 1)
    #> [1] -1.758797
    non_central_chi_squared_cdf(2, 3, 1)
    #> [1] 0.3082525
    non_central_chi_squared_lcdf(2, 3, 1)
    #> [1] -1.176836
    non_central_chi_squared_quantile(0.5, 3, 1)
    #> [1] 3.213009

### Noncentral F Distribution

    # Noncentral F distribution with df1 = 10, df2 = 10 and noncentrality
    # parameter 1
    dist <- non_central_f_distribution(10, 10, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.1142528
    logcdf(dist, 0.5)
    #> [1] -2.169342
    pdf(dist, 0.5)
    #> [1] 0.5754471
    logpdf(dist, 0.5)
    #> [1] -0.552608
    hazard(dist, 0.5)
    #> [1] 0.6496742
    chf(dist, 0.5)
    #> [1] 0.1213237
    mean(dist)
    #> [1] 1.375
    median(dist)
    #> [1] 1.10075
    mode(dist)
    #> [1] 0.7349843
    range(dist)
    #> [1]  0.000000e+00 1.797693e+308
    quantile(dist, 0.2)
    #> [1] 0.6363179
    standard_deviation(dist)
    #> [1] 1.063113
    support(dist)
    #> [1]  0.000000e+00 1.797693e+308
    variance(dist)
    #> [1] 1.130208
    skewness(dist)
    #> [1] 3.609738
    kurtosis(dist)
    #> [1] 51.0825
    kurtosis_excess(dist)
    #> [1] 48.0825

    # Convenience functions
    non_central_f_pdf(1, 5, 2, 1)
    #> [1] 0.3051418
    non_central_f_lpdf(1, 5, 2, 1)
    #> [1] -1.186979
    non_central_f_cdf(1, 5, 2, 1)
    #> [1] 0.3737987
    non_central_f_lcdf(1, 5, 2, 1)
    #> [1] -0.9840377
    non_central_f_quantile(0.5, 5, 2, 1)
    #> [1] 1.507635

### Noncentral T Distribution

    # Noncentral T distribution with 5 degrees of freedom and noncentrality parameter 1
    dist <- non_central_t_distribution(5, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.3021259
    logcdf(dist, 0.5)
    #> [1] -1.196912
    pdf(dist, 0.5)
    #> [1] 0.3360046
    logpdf(dist, 0.5)
    #> [1] -1.090631
    hazard(dist, 0.5)
    #> [1] 0.4814687
    chf(dist, 0.5)
    #> [1] 0.3597165
    mean(dist)
    #> [1] 1.189416
    median(dist)
    #> [1] 1.052851
    mode(dist)
    #> [1] 0.8781834
    range(dist)
    #> [1] -1.797693e+308  1.797693e+308
    quantile(dist, 0.2)
    #> [1] 0.165306
    standard_deviation(dist)
    #> [1] 1.385144
    support(dist)
    #> [1] -1.797693e+308  1.797693e+308
    variance(dist)
    #> [1] 1.918623
    skewness(dist)
    #> [1] 1.26633
    kurtosis(dist)
    #> [1] 13.32067
    kurtosis_excess(dist)
    #> [1] 10.32067

    # Convenience functions
    non_central_t_pdf(0, 5, 1)
    #> [1] 0.2302431
    non_central_t_lpdf(0, 5, 1)
    #> [1] -1.46862
    non_central_t_cdf(0, 5, 1)
    #> [1] 0.1586553
    non_central_t_lcdf(0, 5, 1)
    #> [1] -1.841022
    non_central_t_quantile(0.5, 5, 1)
    #> [1] 1.052851

### Normal Distribution

    # Normal distribution with mean = 0, sd = 1
    dist <- normal_distribution(0, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.6914625
    logcdf(dist, 0.5)
    #> [1] -0.3689464
    pdf(dist, 0.5)
    #> [1] 0.3520653
    logpdf(dist, 0.5)
    #> [1] -1.043939
    hazard(dist, 0.5)
    #> [1] 1.141078
    chf(dist, 0.5)
    #> [1] 1.175912
    mean(dist)
    #> [1] 0
    median(dist)
    #> [1] 0
    mode(dist)
    #> [1] 0
    range(dist)
    #> [1] -Inf  Inf
    quantile(dist, 0.2)
    #> [1] -0.8416212
    standard_deviation(dist)
    #> [1] 1
    support(dist)
    #> [1] -Inf  Inf
    variance(dist)
    #> [1] 1
    skewness(dist)
    #> [1] 0
    kurtosis(dist)
    #> [1] 3
    kurtosis_excess(dist)
    #> [1] 0

    # Convenience functions
    normal_pdf(0)
    #> [1] 0.3989423
    normal_lpdf(0)
    #> [1] -0.9189385
    normal_cdf(0)
    #> [1] 0.5
    normal_lcdf(0)
    #> [1] -0.6931472
    normal_quantile(0.5)
    #> [1] 0

### Pareto Distribution

    # Pareto distribution with scale = 10, shape = 5
    dist <- pareto_distribution(10, 5)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0
    logcdf(dist, 0.5)
    #> [1] -Inf
    pdf(dist, 0.5)
    #> [1] 0
    logpdf(dist, 0.5)
    #> [1] -Inf
    hazard(dist, 0.5)
    #> [1] 0
    chf(dist, 0.5)
    #> [1] 0
    mean(dist)
    #> [1] 12.5
    median(dist)
    #> [1] 11.48698
    mode(dist)
    #> [1] 10
    range(dist)
    #> [1]  0.000000e+00 1.797693e+308
    quantile(dist, 0.2)
    #> [1] 10.4564
    standard_deviation(dist)
    #> [1] 3.227486
    support(dist)
    #> [1]  1.000000e+01 1.797693e+308
    variance(dist)
    #> [1] 10.41667
    skewness(dist)
    #> [1] 4.64758
    kurtosis(dist)
    #> [1] 73.8
    kurtosis_excess(dist)
    #> [1] 70.8

    # Convenience functions
    pareto_pdf(1)
    #> [1] 1
    pareto_lpdf(1)
    #> [1] 0
    pareto_cdf(1)
    #> [1] 0
    pareto_lcdf(1)
    #> [1] -Inf
    pareto_quantile(0.5)
    #> [1] 2

### Poisson Distribution

    # Poisson distribution with lambda = 1
    dist <- poisson_distribution(1)
    # Apply generic functions
    cdf(dist, 5)
    #> [1] 0.9994058
    logcdf(dist, 5)
    #> [1] -0.0005943614
    pdf(dist, 5)
    #> [1] 0.003065662
    logpdf(dist, 5)
    #> [1] -5.787492
    hazard(dist, 5)
    #> [1] 5.159442
    chf(dist, 5)
    #> [1] 7.42832
    mean(dist)
    #> [1] 1
    median(dist)
    #> [1] 1
    mode(dist)
    #> [1] 1
    range(dist)
    #> [1]  0.000000e+00 1.797693e+308
    quantile(dist, 0.2)
    #> [1] 0
    standard_deviation(dist)
    #> [1] 1
    support(dist)
    #> [1]  0.000000e+00 1.797693e+308
    variance(dist)
    #> [1] 1
    skewness(dist)
    #> [1] 1
    kurtosis(dist)
    #> [1] 4
    kurtosis_excess(dist)
    #> [1] 1

    # Convenience functions
    poisson_pdf(0, 1)
    #> [1] 0.3678794
    poisson_lpdf(0, 1)
    #> [1] -1
    poisson_cdf(0, 1)
    #> [1] 0.3678794
    poisson_lcdf(0, 1)
    #> [1] -1
    poisson_quantile(0.5, 1)
    #> [1] 1

### Rayleigh Distribution

    # Rayleigh distribution with sigma = 1
    dist <- rayleigh_distribution(1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.1175031
    logcdf(dist, 0.5)
    #> [1] -2.141291
    pdf(dist, 0.5)
    #> [1] 0.4412485
    logpdf(dist, 0.5)
    #> [1] -0.8181472
    hazard(dist, 0.5)
    #> [1] 0.5
    chf(dist, 0.5)
    #> [1] 0.125
    mean(dist)
    #> [1] 1.253314
    median(dist)
    #> [1] 1.17741
    mode(dist)
    #> [1] 1
    range(dist)
    #> [1]   0 Inf
    quantile(dist, 0.2)
    #> [1] 0.6680472
    standard_deviation(dist)
    #> [1] 0.6551364
    support(dist)
    #> [1]  0.000000e+00 1.797693e+308
    variance(dist)
    #> [1] 0.4292037
    skewness(dist)
    #> [1] 0.6311107
    kurtosis(dist)
    #> [1] 3.245089
    kurtosis_excess(dist)
    #> [1] 0.2450893

    # Convenience functions
    rayleigh_pdf(1)
    #> [1] 0.6065307
    rayleigh_lpdf(1)
    #> [1] -0.5
    rayleigh_cdf(1)
    #> [1] 0.3934693
    rayleigh_lcdf(1)
    #> [1] -0.9327521
    rayleigh_quantile(0.5)
    #> [1] 1.17741

### SaS Point5 Distribution

    # SaS Point5 distribution with location 0 and scale 1
    dist <- saspoint5_distribution(0, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.6686904
    logcdf(dist, 0.5)
    #> [1] -0.402434
    pdf(dist, 0.5)
    #> [1] 0.1707624
    logpdf(dist, 0.5)
    #> [1] -1.767482
    hazard(dist, 0.5)
    #> [1] 0.5154165
    chf(dist, 0.5)
    #> [1] 1.104702
    median(dist)
    #> [1] 0
    mode(dist)
    #> [1] 0
    range(dist)
    #> [1] -Inf  Inf
    quantile(dist, 0.2)
    #> [1] -2.397479
    support(dist)
    #> [1] -Inf  Inf

    # Convenience functions
    saspoint5_pdf(3)
    #> [1] 0.02379919
    saspoint5_lpdf(3)
    #> [1] -3.738104
    saspoint5_cdf(3)
    #> [1] 0.8164545
    saspoint5_lcdf(3)
    #> [1] -0.2027841
    saspoint5_quantile(0.5)
    #> [1] 0

### Skew Normal Distribution

    # Skew Normal distribution with location = 0, scale = 1, shape = 0
    dist <- skew_normal_distribution(0, 1, 0)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.6914625
    logcdf(dist, 0.5)
    #> [1] -0.3689464
    pdf(dist, 0.5)
    #> [1] 0.3520653
    logpdf(dist, 0.5)
    #> [1] -1.043939
    hazard(dist, 0.5)
    #> [1] 1.141078
    chf(dist, 0.5)
    #> [1] 1.175912
    mean(dist)
    #> [1] 0
    median(dist)
    #> [1] 0
    mode(dist)
    #> [1] 0
    range(dist)
    #> [1] -Inf  Inf
    quantile(dist, 0.2)
    #> [1] -0.8416212
    standard_deviation(dist)
    #> [1] 1
    support(dist)
    #> [1] -1.797693e+308  1.797693e+308
    variance(dist)
    #> [1] 1
    skewness(dist)
    #> [1] 0
    kurtosis(dist)
    #> [1] 3
    kurtosis_excess(dist)
    #> [1] 0

    # Convenience functions
    skew_normal_pdf(0)
    #> [1] 0.3989423
    skew_normal_lpdf(0)
    #> [1] -0.9189385
    skew_normal_cdf(0)
    #> [1] 0.5
    skew_normal_lcdf(0)
    #> [1] -0.6931472
    skew_normal_quantile(0.5)
    #> [1] 0

### Student’s T Distribution

    # Student's t distribution with 5 degrees of freedom
    dist <- students_t_distribution(5)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.6808506
    logcdf(dist, 0.5)
    #> [1] -0.3844124
    pdf(dist, 0.5)
    #> [1] 0.3279185
    logpdf(dist, 0.5)
    #> [1] -1.11499
    hazard(dist, 0.5)
    #> [1] 1.027476
    chf(dist, 0.5)
    #> [1] 1.142096
    mean(dist)
    #> [1] 0
    median(dist)
    #> [1] 0
    mode(dist)
    #> [1] 0
    range(dist)
    #> [1] -Inf  Inf
    quantile(dist, 0.2)
    #> [1] -0.9195438
    standard_deviation(dist)
    #> [1] 1.290994
    support(dist)
    #> [1] -Inf  Inf
    variance(dist)
    #> [1] 1.666667
    skewness(dist)
    #> [1] 0
    kurtosis(dist)
    #> [1] 9
    kurtosis_excess(dist)
    #> [1] 6

    # Convenience functions
    students_t_pdf(0, 5)
    #> [1] 0.3796067
    students_t_lpdf(0, 5)
    #> [1] -0.9686196
    students_t_cdf(0, 5)
    #> [1] 0.5
    students_t_lcdf(0, 5)
    #> [1] -0.6931472
    students_t_quantile(0.5, 5)
    #> [1] 0

### Triangular Distribution

    # Triangular distribution with lower = -1, mode = 0, upper = 1
    dist <- triangular_distribution(-1, 0, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.875
    logcdf(dist, 0.5)
    #> [1] -0.1335314
    pdf(dist, 0.5)
    #> [1] 0.5
    logpdf(dist, 0.5)
    #> [1] -0.6931472
    hazard(dist, 0.5)
    #> [1] 4
    chf(dist, 0.5)
    #> [1] 2.079442
    mean(dist)
    #> [1] 0
    median(dist)
    #> [1] 0
    mode(dist)
    #> [1] 0
    range(dist)
    #> [1] -1.797693e+308  1.797693e+308
    quantile(dist, 0.2)
    #> [1] -0.3675445
    standard_deviation(dist)
    #> [1] 0.4082483
    support(dist)
    #> [1] -1  1
    variance(dist)
    #> [1] 0.1666667
    skewness(dist)
    #> [1] 0
    kurtosis(dist)
    #> [1] 2.4
    kurtosis_excess(dist)
    #> [1] -0.6

    # Convenience functions
    triangular_pdf(1)
    #> [1] 0
    triangular_lpdf(1)
    #> [1] -Inf
    triangular_cdf(1)
    #> [1] 1
    triangular_lcdf(1)
    #> [1] 0
    triangular_quantile(0.5)
    #> [1] 0

### Uniform Distribution

    # Uniform distribution with lower = 0, upper = 1
    dist <- uniform_distribution(0, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.5
    logcdf(dist, 0.5)
    #> [1] -0.6931472
    pdf(dist, 0.5)
    #> [1] 1
    logpdf(dist, 0.5)
    #> [1] 0
    hazard(dist, 0.5)
    #> [1] 2
    chf(dist, 0.5)
    #> [1] 0.6931472
    mean(dist)
    #> [1] 0.5
    median(dist)
    #> [1] 0.5
    mode(dist)
    #> [1] 0
    range(dist)
    #> [1] -1.797693e+308  1.797693e+308
    quantile(dist, 0.2)
    #> [1] 0.2
    standard_deviation(dist)
    #> [1] 0.2886751
    support(dist)
    #> [1] 0 1
    variance(dist)
    #> [1] 0.08333333
    skewness(dist)
    #> [1] 0
    kurtosis(dist)
    #> [1] 1.8
    kurtosis_excess(dist)
    #> [1] -1.2

    # Convenience functions
    uniform_pdf(0.5)
    #> [1] 1
    uniform_lpdf(0.5)
    #> [1] 0
    uniform_cdf(0.5)
    #> [1] 0.5
    uniform_lcdf(0.5)
    #> [1] -0.6931472
    uniform_quantile(0.5)
    #> [1] 0.5

### Weibull Distribution

    # Weibull distribution with shape = 1, scale = 1
    dist <- weibull_distribution(1, 1)
    # Apply generic functions
    cdf(dist, 0.5)
    #> [1] 0.3934693
    logcdf(dist, 0.5)
    #> [1] -0.9327521
    pdf(dist, 0.5)
    #> [1] 0.6065307
    logpdf(dist, 0.5)
    #> [1] -0.5
    hazard(dist, 0.5)
    #> [1] 1
    chf(dist, 0.5)
    #> [1] 0.5
    mean(dist)
    #> [1] 1
    median(dist)
    #> [1] 0.6931472
    mode(dist)
    #> [1] 0
    range(dist)
    #> [1]  0.000000e+00 1.797693e+308
    quantile(dist, 0.2)
    #> [1] 0.2231436
    standard_deviation(dist)
    #> [1] 1
    support(dist)
    #> [1] 2.225074e-308 1.797693e+308
    variance(dist)
    #> [1] 1
    skewness(dist)
    #> [1] 2
    kurtosis(dist)
    #> [1] 9
    kurtosis_excess(dist)
    #> [1] 6

    # Convenience functions
    weibull_pdf(1, shape = 1, scale = 1)
    #> [1] 0.3678794
    weibull_lpdf(1, shape = 1, scale = 1)
    #> [1] -1
    weibull_cdf(1, shape = 1, scale = 1)
    #> [1] 0.6321206
    weibull_lcdf(1, shape = 1, scale = 1)
    #> [1] -0.4586751
    weibull_quantile(0.5, shape = 1, scale = 1)
    #> [1] 0.6931472