Why are variance and expected value all we care about?

The expected value and the variance describe the central trend and the spread. These are the minimal quantities one cares about for a first grasp of the data. They are sufficient to define a linear transformation of the data to normalize it (nonlinear transformations aren't so prized).

Higher moments (classically up to order $4$) can be used for a finer description (symmetry and resemblance to a normal law) but they are much less useful, and the tools for their estimation are much less developed.

Also think that the mean and standard deviation are all it takes to describe a Normal distribution, and by the CLT, this distribution is the "destiny" of all samples.

Tags:

Probability Theory

Fourier Analysis

Probability Distributions

Fourier Transform

Probability Limit Theorems

Related

Why do we need the covariant derivative along a curve - why are linear connections not sufficient? What is the difference between a response, output, hidden and latent variables in modeling? Find all natural number $n$ for which $3^9+3^{12}+3^{15}+3^n$ is a perfect cube. Closed form for the series: $\sum_{n=1}^{\infty}x^{n^2}$ Where to go after "olympiad math"? Is $f(a)-f(b) \approx f^{\prime}(b)(a-b)$ for small differences and if so, why? Points in $\Bbb {R}^2$ that can be reached via steps which are $1/5$ of a unit circle. Property of distributions over R x R with identical marginal distributions A binary operation on $G$ satisfies $a\cdot(b\cdot c)=(a\cdot b)\cdot (a\cdot c)$ and has an identity, is it a group? How to extend continous function from $S^1\to S^1$ to $D^2\to D^2$ continously Evaluate $\int_{0}^{\frac{\pi}{4}}\tan xdx $ using idea of Riemann Sum Show that If a sequence is exact the dual sequence is exact

Recent Posts