Looking for a mathematical identity for a summation
Looking for a mathematical identity for a summation
(OP)
Here is the summation I am looking for an identity for:
Σ(nidi^3/N)
I can bring out the N since that is a constant. The ^3 can go away by a variable substituion so I am essentially left with:
ΣXiYi
My question is can I break part that summation so I end up with Xi and Yi in seperate summations/functions?
Some background:
ni and di are two sets of i data points of liquid droplet diameters (di) and % occurance (ni). The summation of the products (nidi^3) represents a numerical average of the droplet size based on volume mean. There is a similar relationship based on the summation of (ni x di) that represents the average droplet size based on a number mean. I am looking to convert the volume mean to number mean without knowing the individual data points.
Because the values of ni would be the same in both cases, if I can isolate the summation of ni from the summation of di I can then divide by the summation of ni and be left with only the summation of di^3. At this point I know of an identity that pulls the exponent out so I can find out what the summation of di would be. The final step would be to reverese the identity I am asking about to re-construct the summation of nidi.
Σ(nidi^3/N)
I can bring out the N since that is a constant. The ^3 can go away by a variable substituion so I am essentially left with:
ΣXiYi
My question is can I break part that summation so I end up with Xi and Yi in seperate summations/functions?
Some background:
ni and di are two sets of i data points of liquid droplet diameters (di) and % occurance (ni). The summation of the products (nidi^3) represents a numerical average of the droplet size based on volume mean. There is a similar relationship based on the summation of (ni x di) that represents the average droplet size based on a number mean. I am looking to convert the volume mean to number mean without knowing the individual data points.
Because the values of ni would be the same in both cases, if I can isolate the summation of ni from the summation of di I can then divide by the summation of ni and be left with only the summation of di^3. At this point I know of an identity that pulls the exponent out so I can find out what the summation of di would be. The final step would be to reverese the identity I am asking about to re-construct the summation of nidi.





RE: Looking for a mathematical identity for a summation
<<A good friend will bail you out of jail, but a true friend
will be sitting beside you saying " Damn that was fun!" - Unknown>>
RE: Looking for a mathematical identity for a summation
you're right about the constant (the sum can be divided instead of dividing each term),
i think i get what you mean with the power, ie making an input of di^3, instead of cubing di (though why you want to or need to i don't know)
but isn't Sum (xiyi) for i =1 to (say)3 ...
x1y1+x2y2+x3y3 ???
RE: Looking for a mathematical identity for a summation
If that's what you're asking, I'm pretty sure that cannot be said. Try it for yourself, with some simple numbers.
V
RE: Looking for a mathematical identity for a summation
I am unclear on what th OP is asking for
<<A good friend will bail you out of jail, but a true friend
will be sitting beside you saying " Damn that was fun!" - Unknown>>
RE: Looking for a mathematical identity for a summation
RE: Looking for a mathematical identity for a summation
- Steve
RE: Looking for a mathematical identity for a summation
In fact, if you know the random distribution of your data (and generally a normal distribution is assumed), there is a probabilistic relationship of the kind you are looking for.
To explain that, let's take a parabolic distribution of data, as the normal distribution is quite cumbersome to handle(but it can be done). Also to make things easier we'll go in reverse from the average of diameters (sizes) to the average of volumes.
So call xi=ni/N the fraction of droplets with diameter di; the summation Σxidi is equivalent to the integral ∫01d(x)dx and similarly for the cube.
If we now assume d(x)=A+B(x-x2) where A may be interpreted as something that's close to an average value (not exactly that, but a kind of) and B, in the same vein, as something close to a variance, we can compute the difference (integrals between 0 and 1):
∫d(x)3dx-(∫d(x)dx)3
This is found to be equal to (if I'm not in error) 2AB2/45+19B3/7560.
Now this quantity will be relatively small if the variance is relatively small with respect to the mean value. Also this expression (or another one computed for a different distribution) can be used to correct the simple equation
davg=3√(Σxidi3)
prex
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RE: Looking for a mathematical identity for a summation
Fe
RE: Looking for a mathematical identity for a summation
RE: Looking for a mathematical identity for a summation
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RE: Looking for a mathematical identity for a summation
the original equation is clearly the mean droplet volume, but you're asking about a "number" mean (as opposed to a "volume" mean) ...
RE: Looking for a mathematical identity for a summation
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RE: Looking for a mathematical identity for a summation
RE: Looking for a mathematical identity for a summation
Assume a general probability density function p(x), not necessarily Gaussian or other, and further assume it to be symmetric about the mean, xo.
then
integral x*p(x)dx = xo def of mean
(1) integral (x-xo)^2*p(x)dx= sigma^2=int(x^2-2xxo+xo^2)*p(x)dx
so that after rearranging and using definition of xo
(2) Int x^2*p(x)dx= sigma^2+xo^2
where sigma is the standard deviation
Now get
integral (x-xo)^3*p(x)dx =integral(x^3-3x^2*xo+3xo^2*x-xo^3)p(x)dx
Now, from the assumption of symmetry, the LHS =0. Rearranging the RHS, yields
integral x^3*p(x)dx= 3xo*(Sigma^2+xo^2)-3xo^3+xo^3
=3xo*(sigma^2+xo^2)
So all you must know is the average dia and its standard deviation to get average Dia^3
Please check the math.
RE: Looking for a mathematical identity for a summation
Found an error at the end . Should read
integral x^3*p(x)dx= 3xo*sigma^2+xo^3
RE: Looking for a mathematical identity for a summation
First, to clarify my question:
The only two pieces of information I am given is the resulting volume-mean dimater of a sample of liquid droplets, D30, and the total number of diamter ranges taken, N. I do not know the individual diameters and occurnece of each diameter range, di and ni.
I do know the general formula for calculating D30 based on a set of di and ni data points is D30 = (Σnidi^3/N)^1/3. Is from 1 to N.
I also know that the general formula for calculating D10, number mean, is D10=Σnidi/N, again i is from 1 to N.
I also can safely assume the data points follow a normal distribution about the mean. This means that values of ni for corresponding values of di will approach a maximum as di approaches D10 from either side.
The post by Prex looks very interesting and I will need to digest that more.
Essentially I am looking for a mathematical way to calculate D10 if I only know D30 and N without any knowledge of what the individual values of ni and di are.
Based on my own work, a simplified summation can be reached where D30/N=ΣAiBi.
Where I am stuck is whether ΣAiBi can be seperated into the respectice variables. I already know that ΣAi +ΣBi does not work and I already know that ΣAi X ΣBi does not work. I am looking to see if there are any identities or work-arounds that split the variables up that I have not found yet.
RE: Looking for a mathematical identity for a summation
The LHS is your D30^3 and the RHS involves D10 (my xo) and the standard deviation.
Case in point -- if sigma =0 then D10=D30 is obvious as seen in the equation.