The triangle inequality, the law of cosines, cauchy inequality and variance of (X-Y)
The triangle inequality, the law of cosines, cauchy inequality and variance of (X-Y).
The law of cosines c^2 = a^2 +b^2 - 2abCos(theta) looks a lot like the variance of (X-Y) where X and Y are random variables and cosine is replaced by covariance...but why?
What is the relationship between triangles and probability of two random variables? Something to do with orthogonal vectors forming the basis for a probability space I'm guessing, but that reaches the limit of what I know about linear algebra. Any insight is appreciated!
Re: The triangle inequality, the law of cosines, cauchy inequality and variance of (X
Basically the angle is related to the correlation between two random variables. If the angle is pi/2 then the random variables are un-correlated and if they are 0 or pi then they are exactly the same or negative.
You can think of the random variables as vectors but instead of looking at orientation, you are looking at correlation in some vector space.
To make sense of this take a look at the formulas for variance, correlation and covariance and compare these to inner products and norms and you'll see the resemblance.
You should construct a vector representation of a random variable and convince yourself geometrically what is going on.