### MOMENTS of a RANDOM VARIABLE:

#### DEFINITION of MOMENTS:

$$\color{blue}{\text{Expected}}$$ value of a $$\color{red}{\text{power}}$$ of a random variable.

We use LOTUS to calculate this expectation: the expected value of a function of the variable (in this case an exponentiation) is obtained by multiplying that function $$\times$$ the pdf, and integrating:

$$\large \mathbb{E}[\color{blue}{X^k}] = \displaystyle\int_{-\infty}^{\infty}\color{blue}{X^k}\,\,\color{green}{\text{pdf}}\,\,\,dx\tag{def. of MOMENT *}$$

$$k$$ is the number of the moment.

For the mean:

$$\large \mathbb{E}[{X^1}] = \displaystyle\int_{-\infty}^{\infty}X^1\,\,\text{pdf}\,\,\,dx$$

There are two types of moments:

1. Raw moment (moment about the origin). Fits perfectly the above definition, of $$\mu'_k=\mathbb{E}\,(X)^k= \mathbb{E}\,(X-0)^k$$. The mean is the first raw moment.

2. Central moment: It is centered around the mean: $$\mu_k=\mathbb{E}\,(X-\mu)^k$$. This is the moment that we need, for instance, to calculate the variance:

Variance is $$\mathrm{Var}[X] = \mathbb{E}\left [ \,(X-\mu)^2 \right ] = \displaystyle\int_{-\infty}^{\infty} (X-\mu)^2 \, \text{pdf}\,dx$$.

Alternatively, it can be defined as the difference between the second and the $$\color{red}{\text{squared}}$$ first raw moments: $$\mathbb{E}[X^2]\,-\,\mathbb{E}[X]^2.$$

1. Mean: First raw moment.
2. Variance: Second central moment.
3. Skew (asymmetry): Third central moment, $$\color{blue}{\text{standardized}}$$ i.e. divided by the $$\sigma^3$$.
4. Kurtosis (peakedness): Fourth central moment, but divided by $$\sigma^3$$, AND subtacting $$-3$$ from the result.

#### METHODS TO OBTAIN MOMENTS:

##### 1. Directly using the definition of Expectation:

We can find $$\mathbb E[X^n]$$ directly and calculate moments through:

$\large \mathbb E[X^n]= \displaystyle \int_{-\infty}^{\infty} x^n f_X(x)\,dx.$

##### 2. Probability generating functions (PGF):

Probability generating functions only work for discrete distributions.

The equation is:

$\large \color{blue}{G(z) = \mathbb E[z^X]= \displaystyle \sum_{x=0}^\infty p_x\, z^x}$

where $$p_x = \Pr\{X=x\}.$$

By differentiating and evaluating at $$1$$ we get factorial moments (not raw moments):

$G_X^{(r)}=\mathbb E[X(X-1)\cdots(X-r+1)]$

##### 3. Moment generating functions (MGF):

$\large M_X(t)=\displaystyle\int_{-\infty}^\infty e^{tx}dF(x)$

Notice that the MGF is the Laplace transform with $$-s$$ replaced by $$t$$:

$$\large \mathscr L\{f\}(s)= \mathbb E [e^{-sX}].$$

They not always exist. The moments are calculated as:

$\mathbb E[X^n]=M_X^{(n)}(0)=\frac{d^nM_X}{dt^n}(0)$

##### 4. Characteristic functions:

They always exist:

$\large \phi(t)=\mathbb E[e^{itX}]= \displaystyle \int_{-\infty}^{\infty} e^{itx}f_X(x)dx$

Notice that the characteristic function is the Fourier Transform of probability density function with the caveat that in probability theory, the characteristic function $$\displaystyle \phi$$ of the probability density function $$\displaystyle f$$ of a random variable $$\displaystyle X$$ of continuous type is defined without a negative sign in the exponential, and since the units of $$\displaystyle x$$ are ignored, there is no $$\displaystyle 2\pi$$ either (from Wikipedia):

$$\large \mathscr F\{f(x)\}=\displaystyle \int_{-\infty}^{\infty}e^{2\pi i k x}\,f(x)\,dx$$

The moments are calculated as:

$\mathbb E[X^k]=(-i)^k\phi_X^{(k)}(0)$