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Probability Distributions

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The Graphing Calculator and Geometry Tool can visualize different types of probability distributions. They graph smooth curves for the probability density functions (PDF) of continuous distributions and plot individual points for the probability mass functions (PMF) of discrete distributions. For cumulative distribution functions (CDFs), they shade areas for continuous distributions and graph segments for discrete distributions.

 

Supported Distributions

Add a distribution by entering the function directly into the expression list or by using the Inference menu.

To open the Inference menu, click Add Item and then select Inference. Or, type inference in an expression line. Then, select Add Distribution. Choose a type of distribution, enter the required inputs, and click Create Distribution to automatically add the function to the expression list.

Screenshot of the Inference Menu.

 

Continuous Distributions

Distribution Inputs Distribution Example Description
Normal distribution Mean and standard deviation normaldist\((0,1)\) Graphs the PDF of a normal distribution with the given mean and standard deviation.

If no standard deviation is provided, it defaults to \(1\). If neither argument is provided, the mean defaults to \(0\).
\(t\)-distribution Degrees of freedom and optional shift and scale inputs tdist\((10)\)

tdist\((10,6,1)\)

 Graphs the PDF of a t-distribution with the given degrees of freedom.

The degrees of freedom must be greater than \(0\).

You can add optional shift and scale values as second and third inputs separated by commas. If no shift value is provided, it defaults to \(0\). If neither argument is provided, the scale defaults to \(1\).
\(\chi^2\) distribution Degrees of freedom chisqdist\((10)\) Graphs the PDF of a chi-square \((\chi^2)\) distribution with the given degrees of freedom.

The degrees of freedom must be greater than \(0\).
Uniform distribution Minimum and maximum uniformdist\((0,1)\) Graphs the PDF of a uniform distribution with the given minimum and maximum.

If no maximum is provided, it defaults to \(1\). If neither argument is provided, the minimum defaults to \(0\).

 

Discrete Distributions

Distribution Inputs Distribution Example Description
Binomial distribution Trials and success probability binomialdist\((10, 0.3)\) Plots the PMF of a binomial distribution with the given number of independent trials and probability of success on each trial.

The number of trials must be a nonnegative integer, and the probability must be \(0\), \(1\), or a number in between.
Poisson distribution Mean poissondist\((5)\) Plots the PMF of a Poisson distribution with the given mean.

The mean must be greater than \(0\).
Geometric distribution Success probability geodist\((0.7)\) Plots the PMF of a geometric distribution with the given success probability.

The probability must be \(0\), \(1\), or a number in between.

 

Cumulative Probability

When you graph a PDF or PMF using any of the previous functions, the Cumulative Probability dropdown menu will appear. Click the menu to calculate cumulative probability and add a visual representation to your model.

For continuous distributions, the shaded area under the curve represents the cumulative probability. For discrete distributions, vertical segments and points represent the cumulative probability.

  • Inner or outer regions: Click Inner to calculate the probability between two bounds or Outer to calculate the probability outside them.
  • Left or right regions:Click Left or Right to calculate probability to the left or right of a specified bound.

Normal Distribution:

Gif opening the Cumulative Probability Footer for the normal distribution, clicking through the different regions, changing the footer inputs, swapping over to bounds, and changing the input.

Binomial Distribution:

Gif opening the Cumulative Probability Footer for the binomial distribution and clicking through the different regions.

 

Default Bounds:

Distribution Inner and Outer Bounds Left and Right Bounds
Normal distribution 1 standard deviation from the mean (approximately \(68\%\) of the total area) The mean
\(t\)-distribution 1 standard deviation from the mean (approximately \(68\%\) of the total area) The mean
Chi-square distribution Range that captures \(50\%\) of the area under the curve, with equal areas on both sides of the peak The median
Uniform distribution Symmetrical interval around the mean (central \(50\%\) of the range) The mean
Discrete distributions
(Binomial, Poisson, and Geometric)		 Range that attempts to capture \(50\%\) of the area symmetrically but is restricted to integer values The median

 

You can calculate the cumulative probability that a random value, \(x\), falls within, outside, to the left, or to the right of specific bounds for both continuous and discrete distributions. Click the computed value to open the Cumulative Probability dropdown and export the value to the expression list.

For continuous distributions, you can also compute the bounds that enclose a desired area (centered around the mean). Click Bounds and then enter the desired area. Click the computed value to open the Upper or Lower Bound dropdown and export the value to the expression list.

Gif of the normal distribution, exporting the cumulative probability, then clicking the Bounds section and exporting the lower and upper bounds.

 

Other Functions to Use with Distributions

Once you’ve entered a distribution, you can apply additional functions. These functions allow you to evaluate or manipulate the distribution in different ways.

Function Try Typing Description
PDF normaldist( ).pdf\((1)\)

normaldist( ).pdf\((x)\)

 Finds the probability density function (PDF) for a continuous distribution or the probability mass function (PMF) for a discrete distribution at a given input.
  • If the input is a number, the calculator returns the height of the PDF or PMF at that value. For a discrete distribution, the calculator will round the input to the nearest integer value before calculating the PMF.
  • If the input contains a free variable, the calculator graphs the PDF or PMF as a function. For a discrete distribution, the .pdf\((x)\) function graphs a step function rather than isolated points.
CDF normaldist( ).cdf\((-1)\)

normaldist( ).cdf\((-1,1)\)

normaldist( ).cdf\((x)\)

 Finds the cumulative distribution function (CDF) at a given input.
  • If the input is a number, the calculator returns the probability that a sample from the distribution, \(x\), is less than or equal to that number (matching the Area output when you click Left in the Cumulative Probability section).
  • If the input is a range, the calculator returns the probability that a random variable, \(x\), falls within that range (matching the Area output when you click Inner in the Cumulative Probability section).
  • If the input contains a free variable, the calculator graphs the CDF as a function.
Inverse CDF normaldist( ).inversecdf\((.25)\)

normaldist( ).inversecdf\((x)\)

 Finds the inverse cumulative distribution function (inverse CDF) at a given input.
  • If the input is a number, the calculator returns the value for which the probability that \(x\) is less than or equal to the input matches the given probability (matching the Bounds output when you click Left in the Cumulative Probability section).
  • If the input contains a free variable, the calculator graphs the inverse CDF as a function.
Mean chisqdist\((10)\).mean Computes the mean of the distribution. The mean of a distribution is equal to the average value of a random sample from the distribution over many trials.
Median chisqdist\((10)\).median Computes the median of the distribution. The median of a distribution is a measure of the central value of the distribution.
Variance chisqdist\((10)\).var Computes the variance of the distribution. The variance of a distribution is a measure of the spread of random samples from the distribution.
Standard deviation chisqdist\((10)\).stdev Computes the standard deviation of the distribution. The standard deviation of a distribution is the square root of the variance of the distribution.
Random normaldist( ).random\((2)\) Generates a list of randomly sampled values from the distribution.
  • The input inside the parentheses specifies how many values the calculator will generate.
  • If the parentheses are empty, the calculator will generate a single random sample.
  • Read more about random values in our Statistics article.

 

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