Re: Probability Density Function

• To: mathgroup at smc.vnet.net
• Subject: [mg74807] Re: [mg74790] Probability Density Function
• From: Darren Glosemeyer <darreng at wolfram.com>
• Date: Fri, 6 Apr 2007 04:18:09 -0400 (EDT)
• References: <200704050812.EAA20924@smc.vnet.net>

```Tony Harris wrote:
> Hello,
>
> I have graphed a normal probability density function with a mean of 10 and a
> standard deviation of 2.5.  I named this function f[x]
>
> I wanted to know how to use the "FilledPlot" to show the area under the
> curve for (x<8) and (7.5<x<11).
>
> I have the probability for x<8 to be .212 and for 7.5<x<11 to be .497 by
> calculating the definite integral of the function f[x] for these two
> probabilities.
>
> I have the cacluation of the probabilities included but no idea on how to
> continue and display each of these in a graph.
>
> Thanks,
>
> T Harris
>
>

This can be accomplished by plotting the pdf and the pdf multiplied by a
Piecewise function which is 1 for the desired region and 0 otherwise and
filling from the Piecewise function to the axis.

This will illustrate the probability that x<8.

FilledPlot[PDF[NormalDistribution[10, 2.5], x]*{1, Piecewise[{{1, x <
8}}]}, {x, 0, 20}, Fills -> {{{2, Axis}, Red}}]

Likewise, this will show the probability for 7.5<x<11.

FilledPlot[PDF[NormalDistribution[10, 2.5], x]*{1, Piecewise[{{1, 7.5 <
x < 11}}]}, {x, 0, 20}, Fills -> {{{2, Axis}, Red}}, PlotPoints -> 20]

The two regions can be shown in the same plot if desired.

FilledPlot[PDF[NormalDistribution[10, 2.5], x]*{1, Piecewise[{{1, x <
8}}], Piecewise[{{1, 7.5 < x < 11}}]}, {x, 0, 20}, Fills -> {{{2, Axis},
Red}, {{3, Axis}, Blue}}, PlotPoints -> 20]

However, there is some overlap in these two regions. A fourth function
for the overlap region, as in the following, could be used to show this.

FilledPlot[PDF[NormalDistribution[10, 2.5], x]*{1, Piecewise[{{1, x < 8}}],
Piecewise[{{1, 7.5 < x < 11}}], Piecewise[{{1, 7.5 < x < 8}}]},
{x, 0, 20},
Fills -> {{{2, Axis}, Red}, {{3, Axis}, Blue}, {{4, Axis},
Purple}}, PlotPoints -> 50]

Darren Glosemeyer
Wolfram Research

```

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