Core Algorithms

Study the Properties of a Custom Probability Distribution

Define a ProbabilityDistribution by specifying its PDF and study the properties of the distribution.

In[1]:=

\[ScriptCapitalD] = ProbabilityDistribution[

1/5/E^((1/5)*Abs[2*x - 3]), {x, -Infinity,

Infinity}]; properties = {{HoldForm[

PDF[\[ScriptCapitalD], x]],

PDF[\[ScriptCapitalD], x]}, {HoldForm[

"Plot the PDF of \[ScriptCapitalD] "],

Plot[PDF[\[ScriptCapitalD], x], {x, -8, 11}, Filling -> Axis,

PlotStyle -> Black,

FillingStyle -> Lighter[Orange, 0.4]]}, {HoldForm[

Integrate[PDF[\[ScriptCapitalD], x], {x, -Infinity, Infinity}]],

Integrate[

PDF[\[ScriptCapitalD], x], {x, -Infinity, Infinity}]}, {HoldForm[

Mean[\[ScriptCapitalD]]],

Mean[\[ScriptCapitalD]]}, {HoldForm[Variance[\[ScriptCapitalD]]],

Variance[\[ScriptCapitalD]]}, {HoldForm[

Skewness[\[ScriptCapitalD]]],

Skewness[\[ScriptCapitalD]]}, {HoldForm[

Kurtosis[\[ScriptCapitalD]]],

Kurtosis[\[ScriptCapitalD]]}, {HoldForm[

Moment[\[ScriptCapitalD], 4]],

Moment[\[ScriptCapitalD], 4]}, {HoldForm[

MomentGeneratingFunction[\[ScriptCapitalD], t]],

MomentGeneratingFunction[\[ScriptCapitalD], t]}};

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