Hi all,
Here is my problem: I have three sizes of widgets, and various amounts of each size. I know the size distribution is normally distributed, with mean m and standard deviation s. Given m and s, I'd like to know the fraction in each size class. I'll call the three sizes D_1, D_2, and D_3, and the fractional amount (of the total) in each class f_1, f_2, and f_3. I think I have three independent equations, as follows:
1 = f_1 + f_2 +f_3
m = f_1*D_1 + f_2*D_2 + f_3*D_3
2*s^2 = (f_1*D_1)^2 + (f_2*D_2)^2 + (f_3*D_3)^2 - 3*m^2
In words, these three equations are:
Sum of the three fractional amounts is one.
Mean is fraction-weighted sum for each size.
Last equation is the equation for variance for a population of three, rearranged. First question: is this logic ok so far?
I think I see I see the path forward...I rearranged equation 1 to solve for f_1:
f_1 = 1 - f_2 - f_3
Then I rearranged eqn. 2 to solve for f_2
f_2 = 1/(D_2) * (-f_1*D_1 - f_3*D_3)
and replaced f_1, so I have:
f_2 = 1/(D_2)*(-(1-f_2-f_3)*D_1 - f_3*D_3)
which I rearranged (hopefully correctly) to get
f_2 = 1/(D_2-D_1) * (-D_1 + f_3*D_1 - f_3*D_3)
Then I try to replace f_1 and f_2 in the third equation, so I have an equation for with only one unknown (f_3). I can show this here (if that would help), but it is fairly hairy looking and I can't solve the remaining equation explicitly for f_3. Neither can my aged version of MathCad. So my next two questions are: am I on the right track? And can any one with better advanced algebra skills come up with a formula for f_3?
I realize this is quadratic...am I correct in assuming there is only one root when m, s, f_1, f_2, and f_3 are all positive?
Side question: is it better to use the phpBB formatting to make the subscripts and superscripts in these equations look good, or to format (as I have tried to do) so that symbolic math readers might have a chance?
Full disclosure: this is an issue that came up in the course of my research for the US Geological Survey...it involves the distribution of small particles of mud and (if it works out) will end up in an existing open-source model for ocean circulation and sediment transport.
Thanks in advance for any help!
Here is my problem: I have three sizes of widgets, and various amounts of each size. I know the size distribution is normally distributed, with mean m and standard deviation s. Given m and s, I'd like to know the fraction in each size class. I'll call the three sizes D_1, D_2, and D_3, and the fractional amount (of the total) in each class f_1, f_2, and f_3. I think I have three independent equations, as follows:
1 = f_1 + f_2 +f_3
m = f_1*D_1 + f_2*D_2 + f_3*D_3
2*s^2 = (f_1*D_1)^2 + (f_2*D_2)^2 + (f_3*D_3)^2 - 3*m^2
In words, these three equations are:
Sum of the three fractional amounts is one.
Mean is fraction-weighted sum for each size.
Last equation is the equation for variance for a population of three, rearranged. First question: is this logic ok so far?
I think I see I see the path forward...I rearranged equation 1 to solve for f_1:
f_1 = 1 - f_2 - f_3
Then I rearranged eqn. 2 to solve for f_2
f_2 = 1/(D_2) * (-f_1*D_1 - f_3*D_3)
and replaced f_1, so I have:
f_2 = 1/(D_2)*(-(1-f_2-f_3)*D_1 - f_3*D_3)
which I rearranged (hopefully correctly) to get
f_2 = 1/(D_2-D_1) * (-D_1 + f_3*D_1 - f_3*D_3)
Then I try to replace f_1 and f_2 in the third equation, so I have an equation for with only one unknown (f_3). I can show this here (if that would help), but it is fairly hairy looking and I can't solve the remaining equation explicitly for f_3. Neither can my aged version of MathCad. So my next two questions are: am I on the right track? And can any one with better advanced algebra skills come up with a formula for f_3?
I realize this is quadratic...am I correct in assuming there is only one root when m, s, f_1, f_2, and f_3 are all positive?
Side question: is it better to use the phpBB formatting to make the subscripts and superscripts in these equations look good, or to format (as I have tried to do) so that symbolic math readers might have a chance?
Full disclosure: this is an issue that came up in the course of my research for the US Geological Survey...it involves the distribution of small particles of mud and (if it works out) will end up in an existing open-source model for ocean circulation and sediment transport.
Thanks in advance for any help!
