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.
Distributions
Continuous Distributions
Distribution Function | Try typing... | Description |
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uniformdist(minimum, 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 \(0\). |
normaldist(mean, standard deviation) | normal\((0,1)\) | 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 \(0\). |
tdist(degrees of freedom) |
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 \(1\). |
chisqdist(degrees of freedom) | chisqdist\((10)\) | Graphs the PDF of a chi-square distribution with the given degrees of freedom.
The degrees of freedom must be greater than \(0\). |
Discrete Distributions
Function | Try typing... | Description |
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poissondist(mean) | poissondist\((5)\) | Plots the PMF of a Poisson distribution with the given mean.
The mean must be greater than \(0\). |
binomialdist(trials, 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. |
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.
Default Bounds:
Distribution | Inner and Outer Bounds | Left and Right Bounds |
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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 |
Uniform distribution | Symmetrical interval around the mean (central \(50\%\) of the range) | 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 |
Poisson distribution | Range that attempts to capture \(50\%\) of the area around the mean symmetrically but is restricted to integer values | The median |
Binomial distribution | Range that attempts to capture \(50\%\) of the area symmetrically but is restricted to integer values | Either the mode or the median (whichever splits the area best) |
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.
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 |
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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.
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CDF |
normaldist( ).cdf\((-1)\)
normaldist( ).cdf\((-1,1)\) normaldist( ).cdf\((x)\) |
Finds the cumulative distribution function (CDF) at a given input.
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Inverse CDF |
normaldist( ).inversecdf\((.25)\)
normaldist( ).inversecdf\((x)\) |
Finds the inverse cumulative distribution function (inverse CDF) at a given input.
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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.
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Learn More
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