# [R] Density function: Area under density plot is not equal to 1. Why?

peter dalgaard pdalgd at gmail.com
Thu Sep 8 20:36:01 CEST 2011

```On Sep 8, 2011, at 19:03 , Albyn Jones wrote:

> Look at
>
>    area <- sum(a\$y)*(a\$x[1]-a\$y[2])
>
> The problem appears to be "a\$x[1]-a\$y[2]"; that is not the length of
> the base of an approximating rectangle, whatever it is :-)

I would assume that that is just a typo for a\$x[1]-a\$x[2], which makes sense if the x grid is regular. The real issue is that from= and to= only restrict the range over which the estimated density is _calculated_ not the one over which it is defined.

I.e. there are bits of the density "sticking out" of the interval from 0 to 1. So the integral is naturally less than 1. To see the point, try (with "a" from the original question).

plot(a)
lines(density(data, from=.4, to=.6), lwd=5)

Greg Snow had the advice how really to estimate a density on a restricted region.

>
> albyn
>
> On Thu, Sep 08, 2011 at 11:36:23AM -0400, Gonçalo Ferraz wrote:
>> Hi, I have a vector 'data' of 58 probability values (bounded between 0 and 1) and want to draw a probability density function of these values. For this, I used the commands:
>>
>> data <- runif(58)
>>
>> a <- density(data, from=0, to=1)
>> plot(a, type="l",lwd=3)
>>
>> But then, when I try to approximate the area under the plotted curve with the command:
>>
>> area <- sum(a\$y)*(a\$x[1]-a\$y[2])
>>
>> I get an area that is clearly smaller than 1.
>>
>> Strangely, if I don't bound the density function with 'to=0,from=1' (which is against my purpose because it extends the pdf beyond the limits of a probability value), I get an area of 1.000. This suggests that I am computing the area well, but using the density function improperly.
>>
>> Why is this happening? Does anyone know how to constrain the density function while still getting a true pdf (summing to 1 under the curve) at the end? Should I use a different function? I read through the density function notes but could not figure out a solution.
>>
>> Thank you!
>>
>> Gonçalo
>>
>> ______________________________________________
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>>
>
> --
> Albyn Jones
> Reed College
> jones at reed.edu
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.

--
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com
"Døden skal tape!" --- Nordahl Grieg

```

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