| Distributions {stats} | R Documentation |
Density, cumulative distribution function, quantile function and random\n", "variate generation for many standard probability distributions are\n", "available in the stats package.\n", "
\n", "\n", "\n", "The functions for the density/mass function, cumulative distribution\n",
"function, quantile function and random variate generation are named in the\n",
"form dxxx, pxxx, qxxx and rxxx respectively.\n",
"
For the beta distribution see dbeta.\n",
"
For the binomial (including Bernoulli) distribution see\n",
"dbinom.\n",
"
For the Cauchy distribution see dcauchy.\n",
"
For the chi-squared distribution see dchisq.\n",
"
For the exponential distribution see dexp.\n",
"
For the F distribution see df.\n",
"
For the gamma distribution see dgamma.\n",
"
For the geometric distribution see dgeom. (This is also\n",
"a special case of the negative binomial.)\n",
"
For the hypergeometric distribution see dhyper.\n",
"
For the log-normal distribution see dlnorm.\n",
"
For the multinomial distribution see dmultinom.\n",
"
For the negative binomial distribution see dnbinom.\n",
"
For the normal distribution see dnorm.\n",
"
For the Poisson distribution see dpois.\n",
"
For the Student's t distribution see dt.\n",
"
For the uniform distribution see dunif.\n",
"
For the Weibull distribution see dweibull.\n",
"
For less common distributions of test statistics see\n",
"pbirthday, dsignrank,\n",
"ptukey and dwilcox (and see the\n",
"‘See Also’ section of cor.test).\n",
"
RNG about random number generation in R.\n",
"
The CRAN task view on distributions,\n", "http://cran.r-project.org/web/views/Distributions.html,\n", "mentioning several CRAN packages for additional distributions.\n", "
\n", "\n", "| Binomial {stats} | R Documentation |
Density, distribution function, quantile function and random\n",
"generation for the binomial distribution with parameters size\n",
"and prob.\n",
"
This is conventionally interpreted as the number of ‘successes’\n",
"in size trials.\n",
"
\n",
"dbinom(x, size, prob, log = FALSE)\n",
"pbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)\n",
"qbinom(p, size, prob, lower.tail = TRUE, log.p = FALSE)\n",
"rbinom(n, size, prob)\n",
"\n",
"\n",
"\n",
"x, q | \n",
"\n",
" vector of quantiles. \n", " |
p | \n",
"\n",
" vector of probabilities. \n", " |
n | \n",
"\n",
" number of observations. If |
size | \n",
"\n",
" number of trials (zero or more). \n", " |
prob | \n",
"\n",
" probability of success on each trial. \n", " |
log, log.p | \n",
"\n",
" logical; if TRUE, probabilities p are given as log(p). \n", " |
lower.tail | \n",
"\n",
" logical; if TRUE (default), probabilities are\n", "P[X ≤ x], otherwise, P[X > x]. \n", " |
The binomial distribution with size = n and\n",
"prob = p has density\n",
"
\n", " p(x) = choose(n, x) p^x (1-p)^(n-x)
\n", "\n", "for x = 0, …, n.\n",
"Note that binomial coefficients can be computed by\n",
"choose in R.\n",
"
If an element of x is not integer, the result of dbinom\n",
"is zero, with a warning.\n",
"
is computed using Loader's algorithm, see the reference below.\n", "
\n", "The quantile is defined as the smallest value x such that\n", "F(x) ≥ p, where F is the distribution function.\n", "
\n", "\n", "\n", "dbinom gives the density, pbinom gives the distribution\n",
"function, qbinom gives the quantile function and rbinom\n",
"generates random deviates.\n",
"
If size is not an integer, NaN is returned.\n",
"
The length of the result is determined by n for\n",
"rbinom, and is the maximum of the lengths of the\n",
"numerical arguments for the other functions. \n",
"
The numerical arguments other than n are recycled to the\n",
"length of the result. Only the first elements of the logical\n",
"arguments are used.\n",
"
For dbinom a saddle-point expansion is used: see\n",
"
Catherine Loader (2000). Fast and Accurate Computation of\n", "Binomial Probabilities; available from\n", "http://www.herine.net/stat/software/dbinom.html.\n", "
\n", "pbinom uses pbeta.\n",
"
qbinom uses the Cornish–Fisher Expansion to include a skewness\n",
"correction to a normal approximation, followed by a search.\n",
"
rbinom (for size < .Machine$integer.max) is based on\n",
"
Kachitvichyanukul, V. and Schmeiser, B. W. (1988)\n", "Binomial random variate generation.\n", "Communications of the ACM, 31, 216–222.\n", "
\n", "For larger values it uses inversion.\n", "
\n", "\n", "\n", "Distributions for other standard distributions, including\n",
"dnbinom for the negative binomial, and\n",
"dpois for the Poisson distribution.\n",
"
\n",
"require(graphics)\n",
"# Compute P(45 < X < 55) for X Binomial(100,0.5)\n",
"sum(dbinom(46:54, 100, 0.5))\n",
"\n",
"## Using \"log = TRUE\" for an extended range :\n",
"n <- 2000\n",
"k <- seq(0, n, by = 20)\n",
"plot (k, dbinom(k, n, pi/10, log = TRUE), type = \"l\", ylab = \"log density\",\n",
" main = \"dbinom(*, log=TRUE) is better than log(dbinom(*))\")\n",
"lines(k, log(dbinom(k, n, pi/10)), col = \"red\", lwd = 2)\n",
"## extreme points are omitted since dbinom gives 0.\n",
"mtext(\"dbinom(k, log=TRUE)\", adj = 0)\n",
"mtext(\"extended range\", adj = 0, line = -1, font = 4)\n",
"mtext(\"log(dbinom(k))\", col = \"red\", adj = 1)\n",
"\n",
"\n",
"