Binomial mass probability function R's dbinom function calculates the proportion of samples (each comprising n observations) expected to contain a given number of 'successes' (y) - assuming those samples are randomly selected from an infinite population, of which a known proportion (P) equal one, and the remainder equal zero. Or, equivalently, this function gives the probability of obtaining f 'successes' in a sample of n successes & failures, where P is the probability a randomly selected observation will be a success. For example: # enter observed frequency y=3 # enter number of observations n=10 # enter probability of success per obs. P=0.1 # find probability of getting y dbinom(y, n, P) Gave: [1] 0.05739563 Note: Because we have not told R to put the results into variables, R has merely printed them out. The frequency and sample size must be whole-numbers and cannot be negative - and P cannot be less than zero, or greater than one. For consistency with its other distribution functions (such as the normal ones) this mass probability function is called dbinom rather than mbinom. If y contained a set of binary values, coded as ones & zeroes (or as TRUE & FALSE) the following code gives the same result as that above. # enter binary data y=c(0,0,1,0,1,1,0,0,0,0) P=0.1 # probability of success per obs. f=sum(y) # find frequency of 1's n=length(y) # find sample size # give probability of getting that result # (assuming the order of 1&0 are unimportant) dbinom(f, n, P) If however, y contains a series of observed frequencies, (but the same sample size) the following code would give the probability of observing each frequency. # enter frequencies f=c(3,2,0,6,9) n=10 # size of samples P=0.1 # probability of success per obs. # give probability of each freq. dbinom(f, n, P) The following code gives the probability for each possible outcome (in ascending order, from 0 to n) as a bargraph. n=4 # size of samples f=0:n # all possible outcomes P=0.3 # probability of success per obs. plot(f, dbinom(f, n, P), type='h')

Cumulative binomial probability function Whereas R's dbinom function yields the binomial probability of observing a frequency (f) in a sample of n values, the pbinom function returns the probability of observing f values or less. For example: # enter sample size n=10 # observed frequency of 1's f=3 # proportion of 1's in population P=0.3 # prob. of getting f or less pbinom(f, n, P) Gave: [1] 0.6496107 Note: Being a cumulative probability, the instruction pbinom(3, 10, .3) produces the same result as sum(dbinom(0:3, 10, .3)) Very often you want to estimate the probability of observing each of a series of frequencies. For instance, pbinom(c(0,1,2,3,4), 4, .3) would give the probability of observing every possible frequency of ones in a sample of (n=) 4 binary observations, assuming a 0.3 probability of selecting a one. It is equivalent to using cumsum(dbinom(0:4,4,0.3)) Of course, pbinom(0, n, P) always yields the same result as dbinom(0, n, P), irrespective of n or P, and pbinom(n, n, P) should always yield 1.

Inverse binomial probability function R's qbinom function does the opposite to its cumulative binomial function, pbinom. In other words, whereas pbinom yields the probability (p) of observing a binomial quantile less than equal to the given value, qbinom gives the quantile at or below which a given proportion of that binomial population lies. For example: # enter proportion <= f p=0.2 # enter sample size n=10 # enter the proportion of 1's in the pop. P=0.3 # give the quantile, f qbinom(p,n,P) Gave: [1] 2 Note: Since frequencies are discrete whole-number values, and any cumulative binomial distribution is therefore a 'step function', different values of p do not necessarily yield different results. For instance both qbinom(.4,10,.3) and qbinom(.6,10,.3) yield 3. For the same reason, unlike normally distributed quantiles the proportion of binomially distributed frequencies which equal or exceed a given quantile (f) is not 1 - qbinom(f, n, .P) but is 1 - qbinom(f,n,.P) + dbinom(f,n,.P)

Random binomial function The rbinom function returns one or more frequencies, randomly selected from an infinite binomially-distributed population of frequencies, of a specified sample size and probability of obtaining a 'success' (P). For example, to select one frequency at random: # enter sample size n=5 # enter proportion of pop=1 P=0.5 # give random binomial freq. rbinom(1,n,P) Gave us: [1] 3 Note: This frequency being randomly selected, repeating these instructions is liable to yield a different outcome. Given P=0.5 the result of executing the above instructions is equivalent to tossing an unbiased coin 5 times, and recording the observed number of 'heads'. If you require a number of such random frequencies, for instance (N=) 50, you could enter rbinom(50, 5, 0.5) This sort of instruction is useful in simulation models. It is equivalent to supplying a set of uniformly-distributed random numbers, between 0 and 1, to the inverse cumulative function. For example as qbinom(runif(10), 5, 0.5) If you obtain many such random frequencies (say N=100000) their distribution can be expected to approach the predicted (binomial) frequency distribution.

Poisson mass probability function R's dpois function returns the probability of observing f 'successes', as predicted by the Poisson model, given a known expected mean frequency (F). For example: # enter frequency in sample f=5 # enter expected frequency F=0.3 # probability of f given F dpois(f, F) Gave: [1] 1.500157e-05 Note: When needed, R prints out, and will accept numbers in exponential format. For example 10000*1e-05 yields 0.1, and 10000000000 yields 1e+10. In other words, in this context, e is an exponent to the base ten. So this e has nothing to do with the base of natural logs, 2.71828 - which is the exponent used to obtain the antilog of natural logs, where x = exp[ln(x)] = eln[x] For graphical purposes it may be useful to give the probability of obtaining a sample containing 0, 1, 2, 3, or 4 ones. For example: f=0:5 # set observable frequencies F=0.3 # set mean frequency # show the prob. of observing each freq. plot(f, dpois(f, F), type='h') Since the Poisson model assumes the (binomial) sample is extremely large, the expected frequency (F) is not the proportion of ones in the population being sampled.

Cumulative Poisson probability function ppois returns the probability of observing a frequency of f or fewer 'successes', as predicted by the Poisson model given a specified mean success rate (F). For example: # enter observed frequency f=5 # enter expected frequency F=3 # give cumulative prob. ppois(f, F) Gave: [1] 0.916082 Note: This instruction produces the same result as adding up the probability of observing a frequency of f or less. If p is the probability of observing f or fewer successes, the probability of obtaining a frequency of greater than f is 1-p. If you need to find the probability of observing f or more, given you expect F, use 1 - ppois(f, F) + dpois(f, F)

Inverse Poisson probability function R's inverse (quantile) Poisson function does the opposite to the cumulative Poisson function. For example: # enter propn <= f p=0.916082 # enter expected frequency F=3 # give quantile (f) qpois(p, F) Gave: [1] 5 Note: Like the other distribution functions on this page, the inverse Poisson function can return the quantiles corresponding to a set of (lower-tailed) probabilities and, if need be, a matching set of expected frequencies - neither of which have to be given in ascending order.

Random Poisson function R's rpois function enables you to obtain a set of (N) frequencies, randomly selected, from a single Poisson distribution (of expected frequency, F). For example: # enter number of frequencies N=5 # enter expected frequency F=10 # randomly select a frequency rpois(N, F) Gave: [1] 14 8 8 7 11 Note: rpois(N, F) is equivalent to these instructions: p=runif(N) qpois(p, F) Where p has a random uniform distribution.