rdoc/randist.rdoc in gsl-1.15.3 vs rdoc/randist.rdoc in gsl-1.16.0.6

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+ new

@@ -1,81 +1,81 @@ # # = Random Number Distributions -# This chapter describes functions for generating random variates and computing -# their probability distributions. Samples from the distributions described -# in this chapter can be obtained using any of the random number generators -# in the library as an underlying source of randomness. +# This chapter describes functions for generating random variates and computing +# their probability distributions. Samples from the distributions described +# in this chapter can be obtained using any of the random number generators +# in the library as an underlying source of randomness. # -# In the simplest cases a non-uniform distribution can be obtained analytically -# from the uniform distribution of a random number generator by applying an +# In the simplest cases a non-uniform distribution can be obtained analytically +# from the uniform distribution of a random number generator by applying an # appropriate transformation. This method uses one call to the random number -# generator. More complicated distributions are created by the +# generator. More complicated distributions are created by the # acceptance-rejection method, which compares the desired distribution against -# a distribution which is similar and known analytically. This usually requires -# several samples from the generator. +# a distribution which is similar and known analytically. This usually requires +# several samples from the generator. # -# The library also provides cumulative distribution functions and inverse -# cumulative distribution functions, sometimes referred to as quantile -# functions. The cumulative distribution functions and their inverses are -# computed separately for the upper and lower tails of the distribution, -# allowing full accuracy to be retained for small results. +# The library also provides cumulative distribution functions and inverse +# cumulative distribution functions, sometimes referred to as quantile +# functions. The cumulative distribution functions and their inverses are +# computed separately for the upper and lower tails of the distribution, +# allowing full accuracy to be retained for small results. # # Contents: -# 1. {Introduction}[link:files/rdoc/randist_rdoc.html#1] -# 1. {The Gaussian Distribution}[link:files/rdoc/randist_rdoc.html#2] -# 1. {The Gaussian Tail Distribution}[link:files/rdoc/randist_rdoc.html#3] +# 1. {Introduction}[link:rdoc/randist_rdoc.html#label-Introduction] +# 1. {The Gaussian Distribution}[link:rdoc/randist_rdoc.html#label-The+Gaussian+Distribution] +# 1. {The Gaussian Tail Distribution}[link:rdoc/randist_rdoc.html#label-The+Gaussian+Tail+Distribution] # ... -# and more, see {the GSL reference}[http://www.gnu.org/software/gsl/manual/"target="_top] -# 1. {Shuffling and Sampling}[link:files/rdoc/randist_rdoc.html#7] +# and more, see {the GSL reference}[https://gnu.org/software/gsl/manual/] +# 1. {Shuffling and Sampling}[link:rdoc/randist_rdoc.html#label-Shuffling+and+Sampling] # -# == {}[link:index.html"name="1] Introduction -# Continuous random number distributions are defined by a probability density -# function, p(x), such that the probability of x occurring in the -# infinitesimal range x to x+dx is p dx. +# == Introduction +# Continuous random number distributions are defined by a probability density +# function, p(x), such that the probability of x occurring in the +# infinitesimal range x to x+dx is p dx. # -# The cumulative distribution function for the lower tail P(x) is defined by the -# integral, and gives the probability of a variate taking a value less than x. +# The cumulative distribution function for the lower tail P(x) is defined by the +# integral, and gives the probability of a variate taking a value less than x. # -# The cumulative distribution function for the upper tail Q(x) is defined by the -# integral, and gives the probability of a variate taking a value greater than -# x. +# The cumulative distribution function for the upper tail Q(x) is defined by the +# integral, and gives the probability of a variate taking a value greater than +# x. # -# The upper and lower cumulative distribution functions are related -# by P(x) + Q(x) = 1 and satisfy 0 <= P(x) <= 1, 0 <= Q(x) <= 1. +# The upper and lower cumulative distribution functions are related +# by P(x) + Q(x) = 1 and satisfy 0 <= P(x) <= 1, 0 <= Q(x) <= 1. # -# The inverse cumulative distributions, x=P^{-1}(P) and x=Q^{-1}(Q) give the -# values of x which correspond to a specific value of P or Q. They can be used -# to find confidence limits from probability values. +# The inverse cumulative distributions, x=P^{-1}(P) and x=Q^{-1}(Q) give the +# values of x which correspond to a specific value of P or Q. They can be used +# to find confidence limits from probability values. # -# For discrete distributions the probability of sampling the integer value k is -# given by p(k), where \sum_k p(k) = 1. The cumulative distribution for the -# lower tail P(k) of a discrete distribution is defined as, where the sum is -# over the allowed range of the distribution less than or equal to k. +# For discrete distributions the probability of sampling the integer value k is +# given by p(k), where \sum_k p(k) = 1. The cumulative distribution for the +# lower tail P(k) of a discrete distribution is defined as, where the sum is +# over the allowed range of the distribution less than or equal to k. # # The cumulative distribution for the upper tail of a discrete distribution Q(k) -# is defined as giving the sum of probabilities for all values greater than k. -# These two definitions satisfy the identity P(k)+Q(k)=1. +# is defined as giving the sum of probabilities for all values greater than k. +# These two definitions satisfy the identity P(k)+Q(k)=1. # -# If the range of the distribution is 1 to n inclusive then P(n)=1, Q(n)=0 -# while P(1) = p(1), Q(1)=1-p(1). +# If the range of the distribution is 1 to n inclusive then P(n)=1, Q(n)=0 +# while P(1) = p(1), Q(1)=1-p(1). # # # -# == {}[link:index.html"name="2] The Gaussian Distribution +# == The Gaussian Distribution # --- # * GSL::Rng#gaussian(sigma = 1) # * GSL::Ran::gaussian(rng, sigma = 1) # * GSL::Rng#ugaussian # * GSL::Ran::ugaussian # -# These return a Gaussian random variate, with mean zero and standard -# deviation <tt>sigma</tt>. +# These return a Gaussian random variate, with mean zero and standard +# deviation <tt>sigma</tt>. # # --- # * GSL::Ran::gaussian_pdf(x, sigma = 1) # -# Computes the probability density p(x) at <tt>x</tt> for a Gaussian distribution +# Computes the probability density p(x) at <tt>x</tt> for a Gaussian distribution # with standard deviation <tt>sigma</tt>. # # --- # * GSL::Rng#gaussian_ratio_method(sigma = 1) # * GSL::Ran::gaussian_ratio_method(rng, sigma = 1) @@ -90,93 +90,93 @@ # * GSL::Cdf::ugaussian_P(x) # * GSL::Cdf::ugaussian_Q(x) # * GSL::Cdf::ugaussian_Pinv(P) # * GSL::Cdf::ugaussian_Qinv(Q) # -# These methods compute the cumulative distribution functions P(x), Q(x) -# and their inverses for the Gaussian distribution with standard +# These methods compute the cumulative distribution functions P(x), Q(x) +# and their inverses for the Gaussian distribution with standard # deviation <tt>sigma</tt>. # -# == {}[link:index.html"name="3] The Gaussian Tail Distribution +# == The Gaussian Tail Distribution # --- # * GSL::Rng#gaussian_tail(a, sigma = 1) # * GSL::Ran#gaussian_tail(rng, a, sigma = 1) # * GSL::Rng#ugaussian_tail(a) # * GSL::Ran#ugaussian_tail(rng) # -# These methods provide random variates from the upper tail of a Gaussian -# distribution with standard deviation <tt>sigma</tt>. +# These methods provide random variates from the upper tail of a Gaussian +# distribution with standard deviation <tt>sigma</tt>. # The values returned are larger than the lower limit <tt>a</tt>, which must be positive. # # --- # * GSL::Ran::gaussian_tail_pdf(x, a, sigma = 1) # * GSL::Ran::ugaussian_tail_pdf(x, a) # -# These methods compute the probability density p(x) at <tt>x</tt> for a Gaussian -# tail distribution with standard deviation <tt>sigma</tt> +# These methods compute the probability density p(x) at <tt>x</tt> for a Gaussian +# tail distribution with standard deviation <tt>sigma</tt> # and lower limit <tt>a</tt>. # -# == {}[link:index.html"name="4] The Bivariate Gaussian Distribution +# == The Bivariate Gaussian Distribution # --- # * GSL::Rng#bivariate_gaussian(sigma_x, sigma_y, rho) # * GSL::Ran::bivariate_gaussian(rng, sigma_x, sigma_y, rho) # -# These methods generate a pair of correlated gaussian variates, -# with mean zero, correlation coefficient <tt>rho</tt> and standard deviations +# These methods generate a pair of correlated gaussian variates, +# with mean zero, correlation coefficient <tt>rho</tt> and standard deviations # <tt>sigma_x</tt> and <tt>sigma_y</tt> in the x and y directions. # # --- # * GSL::Ran::bivariate_gaussian_pdf(x, y, sigma_x, sigma_y, rho) # -# This method computes the probability density p(x,y) at <tt>(x,y)</tt> -# for a bivariate gaussian distribution with standard deviations +# This method computes the probability density p(x,y) at <tt>(x,y)</tt> +# for a bivariate gaussian distribution with standard deviations # <tt>sigma_x, sigma_y</tt> and correlation coefficient <tt>rho</tt>. # -# == {}[link:index.html"name="5] The Exponential Distribution +# == The Exponential Distribution # --- # * GSL::Rng#exponential(mu) # * GSL::Ran::exponential(rng, mu) # -# These methods return a random variate from the exponential +# These methods return a random variate from the exponential # distribution with mean <tt>mu</tt>. # # --- # * GSL::Ran::exponential_pdf(x, mu) # -# This method computes the probability density p(x) at <tt>x</tt> +# This method computes the probability density p(x) at <tt>x</tt> # for an exponential distribution with mean <tt>mu</tt>. # # --- # * GSL::Cdf::exponential_P(x, mu) # * GSL::Cdf::exponential_Q(x, mu) # * GSL::Cdf::exponential_Pinv(P, mu) # * GSL::Cdf::exponential_Qinv(Q, mu) # -# These methods compute the cumulative distribution functions P(x), Q(x) +# These methods compute the cumulative distribution functions P(x), Q(x) # and their inverses for the exponential distribution with mean <tt>mu</tt>. # -# == {}[link:index.html"name="6] The Laplace Distribution +# == The Laplace Distribution # --- # * GSL::Rng#laplace(a) # * GSL::Ran::laplace(rng, a) # -# These methods return a random variate from the Laplace distribution +# These methods return a random variate from the Laplace distribution # with width <tt>a</tt>. # # --- # * GSL::Ran::laplace_pdf(x, a) # -# This method computes the probability density p(x) at <tt>x</tt> +# This method computes the probability density p(x) at <tt>x</tt> # for a Laplace distribution with width <tt>a</tt>. # # --- # * GSL::Cdf::laplace_P(x, a) # * GSL::Cdf::laplace_Q(x, a) # * GSL::Cdf::laplace_Pinv(P, a) # * GSL::Cdf::laplace_Qinv(Q, a) # -# These methods compute the cumulative distribution functions P(x), Q(x) +# These methods compute the cumulative distribution functions P(x), Q(x) # and their inverses for the Laplace distribution with width <tt>a</tt>. # # --- # * GSL::Rng#exppow(a, b) # * GSL::Rng#cauchy(a) @@ -196,38 +196,38 @@ # * GSL::Rng#pareto(a, b) # # # ... # -# and more, see {the GSL reference}[http://www.gnu.org/software/gsl/manual/gsl-ref_19.html#SEC286"target="_top]. +# and more, see {the GSL reference}[https://gnu.org/software/gsl/manual/gsl-ref_19.html#SEC286]. # -# == {}[link:index.html"name="7] Shuffling and Sampling +# == Shuffling and Sampling # --- # * GSL::Rng#shuffle(v, n) # -# This randomly shuffles the order of <tt>n</tt> objects, -# stored in the {GSL::Vector}[link:files/rdoc/vector_rdoc.html] object <tt>v</tt>. +# This randomly shuffles the order of <tt>n</tt> objects, +# stored in the {GSL::Vector}[link:rdoc/vector_rdoc.html] object <tt>v</tt>. # --- # * GSL::Rng#choose(v, k) # -# This returns a {GSL::Vector}[link:files/rdoc/vector_rdoc.html] object with <tt>k</tt> objects -# taken randomly from the {GSL::Vector}[link:files/rdoc/vector_rdoc.html] object <tt>v</tt>. +# This returns a {GSL::Vector}[link:rdoc/vector_rdoc.html] object with <tt>k</tt> objects +# taken randomly from the {GSL::Vector}[link:rdoc/vector_rdoc.html] object <tt>v</tt>. # -# The objects are sampled without replacement, thus each object can only -# appear once in the returned vector. It is required that <tt>k</tt> be less -# than or equal to the length of the vector <tt>v</tt>. +# The objects are sampled without replacement, thus each object can only +# appear once in the returned vector. It is required that <tt>k</tt> be less +# than or equal to the length of the vector <tt>v</tt>. # # --- # * GSL::Rng#sample(v, k) # -# This method is like the method <tt>choose</tt> but samples <tt>k</tt> items -# from the original vector <tt>v</tt> with replacement, so the same object -# can appear more than once in the output sequence. There is no requirement +# This method is like the method <tt>choose</tt> but samples <tt>k</tt> items +# from the original vector <tt>v</tt> with replacement, so the same object +# can appear more than once in the output sequence. There is no requirement # that <tt>k</tt> be less than the length of <tt>v</tt>. # -# {prev}[link:files/rdoc/qrng_rdoc.html] -# {next}[link:files/rdoc/stats_rdoc.html] +# {prev}[link:rdoc/qrng_rdoc.html] +# {next}[link:rdoc/stats_rdoc.html] # -# {Reference index}[link:files/rdoc/ref_rdoc.html] -# {top}[link:files/rdoc/index_rdoc.html] -# +# {Reference index}[link:rdoc/ref_rdoc.html] +# {top}[link:index.html] +# #