Example

We first define a discrete distribution:

Then we can study some properties of this distribution:

$\frac{7}{2}$

$\frac{1}{2}\sqrt{7}$

To return a numeric value, we need to specify the optional parameter numeric or numeric=true.

$1.322875656$

We can set the optional parameter output to output=plot to see a plot demonstration.

To get the formula for computing the specific property of a distribution, we need to specify the optional parameter inert or inert=true.

$\sum _{\mathrm{\_t}\&equals;0}^4\left(\&lcub;\begin{array}{cc}0& \mathrm{\_t}<0\\ \mathrm{binomial}\left(7\&comma;\mathrm{\_t}\right){\left(\frac{1}{2}\right)}^{\mathrm{\_t}}{\left(\frac{1}{2}\right)}^{7-\mathrm{\_t}}& \mathrm{otherwise}\end{array}\right)$

Try another distribution, which is continuous.

$0$

$3$

Example

Say we have observed and recorded some data. We can then put the data onto a list or Vector:

Compute the mode and the 30th percentile of this data sample:

$\left\{2\right\}$

$2$

We can randomly generate a data sample from a known distribution with a specified sample size.

Compare the data sample generated and the original distribution.

$5.21969438283049$

$3.11068660376637$

Then, test the sample to see if it follows the exponential distribution with parameter 5.

Chi-Square Test for Suitable Probability Model

----------------------------------------------
Null Hypothesis:
Sample was drawn from specified probability distribution
Alt. Hypothesis:
Sample was not drawn from specified probability distribution
Bins:                    32
Degrees of freedom:      31
Distribution:            ChiSquare(31)
Computed statistic:      30.784
Computed pvalue:         0.477134
Critical value:          44.9853428040743
Result: [Accepted]
There is no statistical evidence against the null hypothesis

$\left[\mathrm{hypothesis}\&equals;\mathrm{true}\&comma;\mathrm{criticalvalue}\&equals;44.9853428040742\&comma;\mathrm{distribution}\&equals;\mathrm{ChiSquare}\left(31\right)\&comma;\mathrm{pvalue}\&equals;0.477134451984691\&comma;\mathrm{statistic}\&equals;30.78400000\right]$

To read more on different hypothesis tests, you can use the command TestsGuide.

Example

Create a Matrix data sample:

If we want to compute the mean value of this Matrix data sample, then we are going to compute the mean values of three list or Vector data samples stored in the columns correspondingly.

To have both value and plot returned, specify the option output=both.

Example

In our last example, we can use the command ExploreRV to explore some important properties of distributions.

Random Variables: Parameters: $\mathrm{mu\_\_\_R5}$ $\mathrm{sigma\_\_\_R5}$ Statistical Properties: Mean Support Median Variance Mode Moment Generating Function Probability Distribution Function   Cumulative Distribution Function