1.9 Some Special Expectations

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Definition 1.9.1: Mean

Definition

Let

• $X$ : Random variable
• Expectation of $X$ exists

If

$$\mu =E(X)$$

Then we say $\mu$ is the mean value of $X$

Definition 1.9.2: Variance

Definition

Let

• $X$ : Random variable, with
  • Finite mean $\mu$
  • Finite $E[(X-\mu {)}^2]$

If

$${\sigma }^2=\mathrm{Var}(X)=E[(X-\mu {)}^2]$$

Then we say ${\sigma }^2$ is the variance of $X$

Definition: Moments

Definition

In general, if $m$ is a positive integer, and if $M^{(m)}(t)$ means the $m$-th derivative of $M(t)$, we have, by repeated differentiation with respect to $t$, $M^{(m)}(0) = E(X^m)$$

Now $E(X^m) = \int_{-\infty}^\infty x^m f(x),dx \quad \text{or}\quad \sum_{x}x^m p(x),$$ and in mechanics, the integrals (or sums) of this sort are called moments.

Theorem 1.9.1: Constant multiplication and addition with variance

Let

• $X$ : Random variable, with
  • Finite $\mu$
  • Finite ${\sigma }^2$
• $a,b\in \mathbb{R}$ : Constants

Then

$$\mathrm{Var}(aX+b)=a^2\mathrm{Var}(X)$$

Definition 1.9.3: Moment generating function (mgf)

Definition

Let $X$ : Random variable

If

$$\exists h>0\ni E(e^{tX})<\infty ,\forall t\in (-h,h)$$

Then we say $M_X(t)=E(e^{tX})$ is the moment generating function (mgf) of $X$

Theorem 1.9.2: Uniqueness of mgf

Let

• $X,Y$ : Random variables
• $M_X,M_Y$ : mgf of $X,Y$ respectively, existing in some neighborhood of $0$

Then

$$F_X(z)=F_Y(z),\forall z\in \mathbb{R}⟺\exists h>0\ni M_X(t)=M_Y(t),\forall t\in (-h,h)$$

This theorem states that two random variables have the same distribution if and only if they have the same mgf in some neighborhood of zero.

Remark 1.9.1: Characteristic function

Let

• $X$ : Random variable
• $i$ : Imaginary unit
• $t\in \mathbb{R}$

If $\phi (t)=E(e^{itX})$

Then we say $\phi (t)$ is the characteristic function of $X$

Important property of this expectation is that while distributions may not have an mgf, every distribution has a unique characteristic function