# Distributions

The calculator can plot the probability density functions (PDFs), probability mass functions (PMFs), and cumulative distribution functions (CDFs) of several common statistical distributions, as well as compute cumulative probabilities for those distributions.

## Plotting

Each of the following functions will plot a distribution's PDF or PMF.

uniformdist(minimum = 0, maximum = 1)

Plot the PDF of a uniform distribution with the given minimum and maximum. Note that if the second argument is omitted the maximum defaults to 1, and if both arguments are omitted the minimum also defaults to 0.

normaldist(mean = 0, standard deviation = 1)

Plot the PDF of a normal distribution with the given mean and standard deviation. Note that if the second argument is omitted the standard deviation defaults to $$1$$, and if both arguments are omitted the mean also defaults to $$0$$.

tdist(degrees of freedom)

Plot the PDF of a Student's t-distribution with the given degrees of freedom. Note that degrees of freedom must be greater than $$0$$.

poissondist(mean)

Plot the PMF of a Poisson distribution with the given mean. Note that mean must be greater than $$0$$.

binomialdist(trials, probability = 0.5)

Plot the PMF of a binomial distribution given a number of (independent) trials and a probability of success on each trial. Note that trials must be a nonnegative integer and probability must be a number between $$0$$ and $$1$$ (inclusive).

## Computing Cumulative Probabilities

When using any of the above functions to plot a PDF/PMF, a checkbox labeled "Find Cumulative Probability (CDF)" will appear. If that box is checked, the calculator will output the cumulative probability between the values in the "Min" and "Max" input fields. It will also display a visualization of the cumulative probability, either as a shaded region under the curve (for continuous distributions) or as a series of vertical segments and points (for discrete distributions).

## Other functions for use with distributions

The top-level distribution functions offer a simple way to plot PDFs and PMFs and compute cumulative probabilities, but the calculator also provides some functions for working with distribution PDFs/PMFs and CDFs inside of other expressions. Once you have created a distribution, you are able to access its .pdf(), .cdf(), .inversecdf(), and .random() functions. (Learn more about generating random values.) *Note that for discrete distributions there is a difference between what the calculator will plot for the top-level distribution function and what it will plot for the .pdf() function. When using the .pdf() and .cdf() functions, a discrete PMF or CDF will be plotted as a step function rather than as a series of points.

distribution.pdf(value)

Evaluate distribution's PDF/PMF at the given value. If value is numeric, the calculator will output a numeric evaluation. If value is an expression that depends on a free variable, the calculator will plot the PDF/PMF as a function of value.

distribution.cdf(value)

Evaluate distribution's CDF at the given value. If value is numeric, the calculator will output a numeric evaluation. If value is an expression that depends on a free variable, the calculator will plot the CDF as a function of value. For example, normaldist$$(0,1).$$cdf$$(2)$$ will output the probability that a random variable from a standard normal distribution has a value less than or equal to $$2$$.

distribution.cdf(lower, upper)

Compute distribution's cumulative probability between lower and upper. For example, normaldist$$(0,1)$$.cdf$$(-1, 1)$$ will output the probability that a random variable from a standard normal distribution has a value between $$-1$$ and $$1$$.

Note that for discrete distributions $$d.$$pdf$$(x)$$ will round $$x$$ to the nearest integer, and a plot of d.pdf(x) will look like a piecewise-constant function. To plot a set of points instead, you could use a table or a point list: $$R=[0…10], (R, d.$$pdf$$(R))$$. The .pdf() and .cdf() functions let you combine distributions in interesting ways. For example, by plotting the difference between their PDFs, it's possible to see that a t-distribution approaches a standard normal distribution as its number of degrees of freedom increases:

distribution.inversecdf(value)

Compute distribution's inverse cumulative density at value. If value is numeric, the calculator will output a numeric evaluation. If value is an expression that depends on a free variable, the calculator will plot the inverse CDF as a function of value. For example, normaldist$$(0,1)$$.inversecdf$$(0.5)$$ will output $$0$$ because normaldist$$(0,1).$$cdf$$(0)$$ is $$0.5$$.