Mean volume of sphere (normal distributed radius)

In summary: The mean of the first set is about 9, while the mean of the second set is 8. This is because the larger numbers have a greater weight in the calculation due to the exponent. Similarly, in the integral for the mean volume, the larger radii have a greater weight due to the 3rd power dependence in the volume function. This leads to a larger mean volume compared to the mean radius.
  • #1
ChrisVer
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I read in http://www-library.desy.de/preparch/books/vstatmp_engl.pdf page 29 (43 for pdf) that the mean volume is:
[itex]<V> = \int_{-\infty}^\infty dr V(r) N(r| r_0,s)[/itex]
I have two questions.
Q1: why do they take the radius to be from -infinity to +infinity and not from 0 to infinity?
Q2: is there an intuitive way to see/describe why the mean volume is larger than the one obtained from the mean radius?

Thanks
 
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  • #2
It would appear the normal distribution about ## r=r_o ## is an approximation they want you to use even though that allows for a finite probability that ## r<0 ##. The integral from minus infinity to plus infinity can be solved in closed form, but I don't believe you get a closed form if the limits were zero to infinity even though the answer would be nearly identical. The ## V=(4/3) \pi r^3 ## function will add more weight to the larger r's because of the 3rd power dependence, making the mean volume larger than ## V=(4/3) \pi r_o^3 ##, but I don't have a simple intuitive description for that.
 
  • #3
ChrisVer said:
Q2: is there an intuitive way to see/describe why the mean volume is larger than the one obtained from the mean radius?

You could compare the mean of ##\{1^3, 2^3, 3^3\} ## with ##2^3##.
 
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1. What is the formula for calculating the mean volume of a sphere with a normal distributed radius?

The formula for calculating the mean volume of a sphere with a normal distributed radius is V = (4/3)πμ3, where μ is the mean radius.

2. How is the mean radius of a sphere determined in a normal distribution?

In a normal distribution, the mean radius of a sphere is determined by taking the average of all the possible radius values within the distribution.

3. Can the mean volume of a sphere with a normal distributed radius be negative?

No, the mean volume of a sphere with a normal distributed radius cannot be negative. It is a measure of central tendency and represents the average volume of all the spheres within the distribution.

4. Is the mean volume of a sphere with a normal distributed radius affected by outliers?

Yes, the mean volume of a sphere with a normal distributed radius can be affected by outliers. Outliers are extreme values that can skew the data and therefore, impact the calculated mean volume.

5. How is the mean volume of a sphere with a normal distributed radius used in scientific research?

The mean volume of a sphere with a normal distributed radius is commonly used in scientific research as a measure of central tendency for data involving spheres. It can also be used to compare different distributions and make statistical inferences.

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