<selectRandomNumbers/>

<selectRandomNumbers/> is a Math component that generates document variants by selecting random numbers from a specified range of values, from a normal, log-normal, or mixture-of-normals distribution, or from one of the discrete count distributions (hypergeometric, binomial, and Poisson).

The numbers are selected once for each variant and stay the same while the activity is open, so <selectRandomNumbers/> has no resample action. For values a reader can redraw, use <sampleRandomNumbers/> instead.

Attributes and Properties

Examples

Example: Default <selectRandomNumbers/>

Without any attributes specified, the default selection type is uniform, and the default interval is $[0, 1)$.

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Example: Assign descriptive names to selections

Three numbers are selected. Selections can be assigned more descriptive names by defining individual components which reference the selected values.

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Attribute Examples

Attribute Example: numToSelect

The <selectRandomNumbers/> component is used with the numToSelect attribute to select multiple random values in the default interval $[0,1)$. The default type is uniform.

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Attribute Example: from, to

The <selectRandomNumbers/> component is used with the from and the to attributes to modify the default selection interval. The default type is uniform.

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Attribute Example: type = “discreteUniform”

The type attribute of the <selectRandomNumbers/> component is specified as discreteUniform. The discreteUniform selection generates values from an evenly spaced range defined by the attributes to, from, and step. If left unspecified, the default step is 1.

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Attribute Example: exclude

The type attribute of the <selectRandomNumbers/> component is specified as discreteUniform. Values can be excluded from the specified range used with this type using the exclude attribute.

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Attribute Example: step

The type attribute of the <selectRandomNumbers/> component is specified as discreteUniform. The discreteUniform selection generates values from an evenly spaced range defined by the attributes to, from, and step. If left unspecified, the default step is 1.

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Attribute Example: type = “gaussian”

The type attribute of the <selectRandomNumbers/> component is specified as gaussian. The gaussian selection generates standard normal variables (mean 0, variance 1). Alternatively, the mean as well as the variance or the standardDeviation attributes may be defined.

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Attribute Example: gaussian with mean and variance

The type attribute of the <selectRandomNumbers/> component is specified as gaussian. The gaussian selection generates standard normal variables (mean 0, variance 1). In this example, the mean and the variance are defined with attributes.

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Attribute Example: gaussian with mean and standardDeviation

The type attribute of the <selectRandomNumbers/> component is specified as gaussian. The gaussian selection generates standard normal variables (mean 0, variance 1). In this example, the mean and the standardDeviation are defined with attributes.

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Attribute Example: type = “logNormal”

The logNormal type raises $e$ to the power of a normal variable, so every value is positive and the distribution is skewed to the right — the usual shape for quantities such as incomes, particle sizes, or times to finish a task.

Its parameters describe the normal distribution being exponentiated, not the values themselves, which is why they are named logMean and logStandardDeviation rather than mean and standardDeviation. Either logStandardDeviation or logVariance may be given, as with the gaussian type. If unspecified, logMean is 0 and logStandardDeviation is 1.

Writing $\mu$ for logMean and $\sigma$ for logStandardDeviation, the reported mean is $e^{\mu + \sigma^2/2}$ and the reported variance is $(e^{\sigma^2} - 1)\,e^{2\mu + \sigma^2}$. Both grow very fast with $\sigma$, so a large enough one reports Infinity for moments beyond what a number can hold, and a value too large to represent comes back as Infinity as well. A value too small to represent rounds to 0, which is the only way one can fail to be strictly positive.

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Attribute Example: type = “normalMixture”

The normalMixture type draws each value from one of several normal distributions: a component is chosen at random, and the value comes from that component alone. It is the usual way to describe a population made of distinct groups, and the usual way to produce a bimodal set of values, which no single gaussian can.

The components are described by three lists. means gives each component’s center, and how many are listed is how many components there are — it is the one of the three with no default. standardDeviations (or variances, as with the gaussian type) gives each component’s spread, and weights gives each component’s share of the values. Weights are relative and need not add up to 1: weights="1 2" above means a third of the values come from the first component and two thirds from the second. A list holding a single value applies to every component, which is how the defaults — a spread of 1, and equal weights — are written.

The reported mean is the weighted average of the component means. The reported variance is the weighted average of the component variances plus a second term for how far the components sit from one another. That second term is zero exactly when every component carrying weight shares one center — a component whose weights entry is 0 contributes to neither term — and is positive otherwise. Writing $p_i$, $\mu_i$ and $\sigma_i$ for the proportion, mean and standard deviation of each component, it is $\sum_i p_i \sigma_i^2 + \sum_i p_i (\mu_i - \mu)^2$, where $\mu$ is the reported mean.

Whether the values come out looking bimodal depends on the components. Two equally weighted components of equal spread separate into two visible humps only once their centers are more than $2\sigma$ apart.

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Attribute Example: type = “hypergeometric”

The hypergeometric type counts the successes obtained when drawing numDraws items without replacement from a population of numTotal items containing numSuccesses successes. It is the without-replacement counterpart of the binomial type.

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Attribute Example: type = “binomial”

The binomial type counts the successes in numTrials independent trials that each succeed with the given probability. If unspecified, numTrials is 1 and probability is 0.5.

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Attribute Example: type = “poisson”

The poisson type is determined entirely by its mean, which also equals its variance. Unlike the gaussian type, whose mean defaults to 0, the mean of a poisson distribution defaults to 1, since a Poisson distribution with mean 0 always returns 0.

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Limits on the discrete distributions

The three discrete distributions — hypergeometric, binomial, and poisson — draw their values one item, trial, or event at a time, so very large parameters would take a long time and leave the page unresponsive while they ran. Parameters that would need more than ten million draws for a single value are therefore treated as unusable, giving NaN values and statistics along with a warning explaining what to change. The limit is far above the populations, trial counts, and rates that arise in practice; it is there so that mistyping an extra digit reports a problem instead of freezing the activity. An order of magnitude below that, from a million draws per value, the values are still selected as asked, but a warning notes that selection may be slow. Note that both figures apply to each value individually — a large numToSelect still takes proportionally longer.

hypergeometric also has no defaults to fall back on. If any of numTotal, numSuccesses, or numDraws is omitted, or they do not describe a population, every value is NaN, as are the reported mean, variance, and standardDeviation.