Distributions for random variables

These functions describe how a random variable is distributed, for use in rand_var(). Each takes the same parameters, in the same order, as R's corresponding random number function:

Usage

distribution(head, ...)

normal(mean, sd)

uniform(min, max)

exponential(rate)

bernoulli(prob)

Arguments

head

The name under which the service knows the distribution.

...

The distribution's parameters, each a number, a numeric vector, or an expression without random variables.

mean, sd

The mean and the standard deviation, as in rnorm().

min, max

The smallest and the largest value, as in runif().

rate

The rate, as in rexp(); the mean is 1 / rate.

prob

The probability of a 1, as in rbinom(n, 1, prob).

Value

A distribution, to be passed to rand_var() or set_distribution().

Details

• normal(mean, sd), like rnorm(): the mean and the standard deviation.

• uniform(min, max), like runif(): the smallest and the largest value.

• exponential(rate), like rexp(): the rate, so that the mean is 1 / rate.

• bernoulli(prob), like rbinom(n, 1, prob): 1 with probability prob, and 0 otherwise.

For any other distribution, draw a sample in R and pass it to empirical(). distribution(head, ...) names a distribution that the service offers but this package has no function for yet; the service refuses a name it does not know.

Parameters that depend on a decision

A parameter may be an expression of decision variables rather than a number. The distribution then depends on the decision: a demand whose mean falls as the price rises, or a breakdown that becomes less likely the more is spent on maintenance. A parameter cannot contain a random variable.

A parameter given as a number is checked right away: a rate that is not positive, a probability outside 0 to 1, or a min above max is an error. A parameter given as an expression cannot be checked in advance, so give the decision variables in it bounds that keep it in range.

Vector parameters

A parameter may be a vector. The distribution then describes that many random variables, drawn independently: normal(c(6, 5, 4), 1) is three normal variables with means 6, 5 and 4 and standard deviation 1. Each parameter has length 1 or the common length.

Examples

m <- model()
lead_time <- rand_var(m, "lead_time", uniform(2, 5))      # between 2 and 5 days
gap       <- rand_var(m, "gap", exponential(1 / 30))      # 30 minutes on average
fails     <- rand_var(m, "fails", bernoulli(0.02))        # 1 with probability 0.02

# the more is spent on maintenance, the less likely a breakdown
spend  <- num_var(m, "spend", lower = 0, upper = 10)
breaks <- rand_var(m, "breaks", bernoulli(0.2 - 0.015 * spend))