What is Gauss-Markov theorem? Why is it so important?

Expert Solution

I would start by talking about a scatter plot. This a plot of data along an x and y axis. For example, x could be years and y could be height and we could be looking at a scatter plot of people's height vs how old they are. Now some people are tall and others are short, but somewhere in there you have an average across all people of a given age. In fact, the average (or mean) of the data as we sweep across different ages forms a line in our scatter plot. The Gauss-Markov theorem states that if people generally have about the same variance in height at each age, then the "ordinary least squares" estimate of the mean (the slope and offset of the average line) would give you the best unbiased estimate of how people's height varies with age on average.

Gauss-Bonnet theorem related the topology of a manifold to its geometry. It is an extraordinary result which expresses the total (Gaussian) curvature of a compact manifold in terms of its Euler characteristic (a topological invariant).

If you 'deform' a sphere, its curvature in the vicinity of the point of deformation has changed. But the Euler characteristic, being a topological invariant will stay same. What Guess-Bonnet theorem tells us that the total curvature of the deformed sphere is equal of the total curvature of the original sphere (4π in this case).

Rahul Sunny answered 2 years ago

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