Expectation, and moments of a random variable

Suppose a random variable X, from which N samples are known $(X_{1}, X_{2}, X_{3}, . . ., X_{n})$ . Expectation is defined with the following expression:

E [x] = \lim_{N \to \infty} \frac{X_{1} + X_{2} + X_{3} + X_{N}}{N} = \int_{- \infty}^{\infty} x p_{X} (x) d x

Moments

For any positive integer r, the r-th moment of the random variable characterized by the PDF $p_{x} (X)$ are given by the following expression:

μ_{r} = E [x^{r}] = \int_{- \infty}^{\infty} x^{r} p_{X} (x) d x

For discrete random variables, the moments are calculated as:

μ_{r} = E [x^{r}] = \sum_{i = 1}^{n} x_{t}^{r} P (X = x_{i})

The first moment defines the mean of the random variable ( $μ$ ), while the second, the mean squared value.

Moments can be defined about a given number, for example, the mean. Moments about the mean are known as central moments, and calculated as follows:

μ_{0 r} = E [(x - μ)^{r}] = \int_{- \infty}^{\infty} (x - μ)^{r} p_{X} (x) d x

For discrete random variables, the central moments are calculated as follows:

μ_{0 r} = \sum_{i = 1}^{n} (x_{i} - μ)^{r} P (X = x_{i})

The Variance $σ_{x^{2}}$ is a measure of dispersion around the mean value. It is the central moment for r=2.

σ_{x}^{2} = E [{(x - μ)}^{2}] = \int_{- \infty}^{\infty} (x - μ)^{2} p_{X} (x) d x

In the case of discrete random variables, it is calculated as follows:

σ_{x}^{2} = \sum_{n = - 1}^{N} (x_{i} - μ)^{2} P (X = x_{i})

Review of Basic Concepts of Probability

Expectation, and moments of a random variable

Moments