Here, we discuss probability distributions functions in R, setting parameters, getting random samples, density or mass, cumulative probability and quantile.
The table below contains 16 probability distributions from the "stats" package in R. It contains information on how to sample from each distribution with different parameter values, derive probability density or mass for any \(x\) value, cumulative probability for any quantile (\(q\)) value, and deriving quantile values for any probability (\(p\)).
| Distribution | Random Sample of Length \(n\) | Probability Density or Mass at \(x\) | Cumulative Probability at \(q\) | Quantile for \(p\) |
|---|---|---|---|---|
| Normal Distribution | rnorm(n, mean=0, sd=1) | dnorm(x, mean=0, sd=1) | pnorm(q, mean=0, sd=1) | qnorm(p, mean=0, sd=1) |
| Student’s t Distribution | rt(n, df, ncp) | dt(x, df, ncp) | pt(q, df, ncp) | qt(p, df, ncp) |
| Binomial Distribution | rbinom(n, size, prob) | dbinom(x, size, prob) | pbinom(q, size, prob) | qbinom(p, size, prob) |
| Poisson Distribution | rpois(n, lambda) | dpois(x, lambda) | ppois(q, lambda) | qpois(p, lambda) |
| Exponential Distribution | rexp(n, rate=1) | dexp(x, rate=1) | pexp(q, rate=1) | qexp(p, rate=1) |
| Uniform Distribution | runif(n, min=0, max=1) | dunif(x, min=0, max=1) | punif(q, min=0, max=1) | qunif(p, min=0, max=1) |
| Chi-squared Distribution | rchisq(n, df, ncp=0) | dchisq(x, df, ncp=0) | pchisq(q, df, ncp=0) | qchisq(p, df, ncp=0) |
| F Distribution | rf(n, df1, df2, ncp) | df(x, df1, df2, ncp) | pf(q, df1, df2, ncp) | qf(p, df1, df2, ncp) |
| Lognormal Distribution | rlnorm(n, meanlog=0, sdlog=1) | dlnorm(x, meanlog=0, sdlog=1) | plnorm(q, meanlog=0, sdlog=1) | qlnorm(p, meanlog=0, sdlog=1) |
| Negative Binomial Distribution | rnbinom(n, size, prob, mu) | dnbinom(x, size, prob, mu) | pnbinom(q, size, prob, mu) | qnbinom(p, size, prob, mu) |
| Geometric Distribution | rgeom(n, prob) | dgeom(x, prob) | pgeom(q, prob) | qgeom(p, prob) |
| Hypergeometric Distribution | rhyper(nn, m, n, k) | dhyper(x, m, n, k) | phyper(q, m, n, k) | qhyper(p, m, n, k) |
| Beta Distribution | rbeta(n, shape1, shape2, ncp=0) | dbeta(x, shape1, shape2, ncp=0) | pbeta(q, shape1, shape2, ncp=0) | qbeta(p, shape1, shape2, ncp=0) |
| Gamma Distribution | rgamma(n, shape, rate=1, scale=1/rate) | dgamma(x, shape, rate=1, scale=1/rate) | pgamma(q, shape, rate=1, scale=1/rate) | qgamma(p, shape, rate=1, scale=1/rate) |
| Cauchy Distribution | rcauchy(n, location=0, scale=1) | dcauchy(x, location=0, scale=1) | pcauchy(q, location=0, scale=1) | qcauchy(p, location=0, scale=1) |
| Weibull Distribution | rweibull(n, shape, scale=1) | dweibull(x, shape, scale=1) | pweibull(q, shape, scale=1) | qweibull(p, shape, scale=1) |
When a parameter’s value in a function is pre-specified to a default value, it does not need to be specified, unless it is to be set to a different parameter value.
For example, for pnorm(1.96, mean=0, sd=1), the
following are the same:
[1] 0.9750021or:
[1] 0.9750021and:
[1] 0.9750021Note that the order of 0 and 1 matters as pnorm(1.96, 1)
and pnorm(1.96, 1, 0) will give different results.
Similarly for dexp(x, rate=1), the following are
the same:
[1] 0.006737947and:
[1] 0.006737947Sample 10 observations from the binomial distribution with \(\tt{size} = 12\) and \(p = 0.7\):
[1] 10 8 8 8 7 8 12 10 8 8Calculate the density at \(x=0.7\), in the uniform distribution with \(\min = 0.5\) and \(\max = 1\):
[1] 2Calculate the cumulative density at \(x=2.2\), in the Student’s t distribution with degrees of freedom \(\tt{df} = 12\):
[1] 0.9759316Derive the quantile for \(p = 0.95\), in the F distribution with degrees of freedom, \(\tt{df1} = 1\) and \(\tt{df2} = 25\):
[1] 4.241699The feedback form is a Google form but it does not collect any personal information.
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