In integral geometry, (otherwise called geometric probability theory) Hadwiger's theorem states that the space of translation-invariant, finitely additive, not-necessarily-nonnegative set functions defined on finite unions of compactconvex sets in Rn consists (up toscalar multiples) of one "measure" that is "homogeneous of degree k" for each k = 0, 1, 2, ..., n, and linear combinations of those "measures".
"Homogeneous of degree k" means that rescaling any set by any factor c > 0 multiplies the set's "measure" by ck. The one that is homogeneous of degree n is the ordinary n-dimensional volume. The one that is homogeneous of degree n − 1 is the "surface volume." The one that is homogeneous of degree 1 is a mysterious function called the mean width, a misnomer. The one that is homogeneous of degree 0 is the Euler characteristic.
The theorem was proved by Hugo Hadwiger, and led to further work on intrinsic volumes.
References
An account and a proof of Hadwiger's theorem may be found in Introduction to Geometric Probability by Daniel Klain and Gian-Carlo Rota, Cambridge University Press, 1997.
The sample mean is a point estimate of the population mean, i.e, it is a single value which we use to represent the population mean.
However, the sample mean varies in repeated samples from the population and thus we need to assess (probabilistically) how close the sample mean is to the population mean.
The confidence intervals enclosing the true population mean (denoted as a red vertical line) are drawn in blue, whereas the intervals not containing the population mean are drawn in yellow.
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