R help - Nov 2004 - density estimation: compute sum(value * probability) for given distribution

Dear R users,

This is a KDE beginner's question. 
I have this distribution:
> length(cap)
[1] 200
> summary(cap)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  459.9   802.3   991.6  1066.0  1242.0  2382.0 
I need to compute the sum of the values times their probability of
occurence.

The graph is fine,
den <- density(cap, from=min(cap), 
       to=max(cap), give.Rkern=F)
plot(den)

However, how do I compute sum(values*probabilities)? The
probabilities produced by the density function sum to only 26%:
> sum(den$y)
[1] 0.2611142

Would it perhaps be ok to simply do
> sum(den$x*den$y) * (1/sum(den$y))
[1] 1073.22
?

Thank you,
b.

First thing you probably should realize is that density is _not_
probability.  A probability density function _integrates_ to one, not _sum_
to one.  If X is an absolutely continuous RV with density f, then Pr(X=x)=0
for all x, and Pr(a < X < b) = \int_a^b f(x) dx.

sum x*Pr(X=x) (over all possible values of x) for a discrete distribution is
just the expectation, or mean, of the distribution.  The expectation for a
continuous distribution is \int x f(x) dx, where the integral is over the
support of f.  This is all elementary math stat that you can find in any
textbook.

Could you tell us exactly what you are trying to compute, or why you're
computing it?

HTH,
Andy
> From: bogdan romocea
> 
> Dear R users,
> 
> This is a KDE beginner's question. 
> I have this distribution:
> > length(cap)
> [1] 200
> > summary(cap)
>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
>   459.9   802.3   991.6  1066.0  1242.0  2382.0 
> I need to compute the sum of the values times their probability of
> occurence.
> 
> The graph is fine,
> den <- density(cap, from=min(cap), 
>        to=max(cap), give.Rkern=F)
> plot(den)
> 
> However, how do I compute sum(values*probabilities)? The
> probabilities produced by the density function sum to only 26%: 
> > sum(den$y)
> [1] 0.2611142
> 
> Would it perhaps be ok to simply do
> > sum(den$x*den$y) * (1/sum(den$y))
> [1] 1073.22
> ?
> 
> Thank you,
> b.
> 
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide! 
> http://www.R-project.org/posting-guide.html
> 
>

bogdan romocea wrote:
> Dear R users,
> 
> This is a KDE beginner's question. 
> I have this distribution:
> 
>>length(cap)
> 
> [1] 200
> 
>>summary(cap)
> 
>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
>   459.9   802.3   991.6  1066.0  1242.0  2382.0 
> I need to compute the sum of the values times their probability of
> occurence.
> 
> The graph is fine,
> den <- density(cap, from=min(cap), 
>        to=max(cap), give.Rkern=F)
> plot(den)
> 
> However, how do I compute sum(values*probabilities)? 
I don't get the point. You are estimating using a gaussian kernel.
Hint: What's the probability to get x=0 for a N(0,1) distribution?
So sum(values*probabilities) is zero!

 > The
> probabilities produced by the density function sum to only 26%: 
and could also sum to, e.g., 783453.9, depending on the number of 
observations and the estimated parameters of the desnity ...
>>sum(den$y)
> 
> [1] 0.2611142
> 
> Would it perhaps be ok to simply do
> 
>>sum(den$x*den$y) * (1/sum(den$y))
> 
> [1] 1073.22
> ?
No. den$x is a point where the density function is equal to den$y, but 
den$y is not the probability to get den$x (you know, the stuff with 
intervals)! I fear you are mixing theory from discrete with continuous 
distributions.

Uwe Ligges


> Thank you,
> b.
> 
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide!
http://www.R-project.org/posting-guide.html

On 13-Nov-04 bogdan romocea wrote:
> Dear R users,
> 
> However, how do I compute sum(values*probabilities)? The
> probabilities produced by the density function sum to only 26%: 
>> sum(den$y)
> [1] 0.2611142
> 
> Would it perhaps be ok to simply do
>> sum(den$x*den$y) * (1/sum(den$y))
> [1] 1073.22
> ?
What you're missing is the "dx"! A density estimation estimates
the probability density function g(x) such that int[g(x)*dx] = 1,
and R's 'density' function returns estimated values of "g"
at a
discrete set of points.

An integral can be approximated by a discrete summation of the
form

    sum(g(x.i)*delta.x

You can recover the set of x-values at which the density is estimated,
and hence the implicit value of delta.x, from the returned density.

Example:

  X<-rnorm(1000)
  f<-density(X)
  x<-f$x
  delta.x<-x[2]-x[1]
  g<-f$y
  sum(g*delta.x)

  [1] 1.000976

Hoping this helps,
Ted.


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Date: 14-Nov-04                                       Time: 08:50:53
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