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.
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.

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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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