Expectation value (mean value), mean square deviation

Definition 4 (Expectation value)   Let $x$ be a random variable with probablity density $\rho(x)$. Then,

$$\fbox{$ \displaystyle \bar{x} \equiv \langle x \rangle \equiv \int_{-\infty}^{\infty}dx \rho(x)x $}$$ (32)

is called expectation value (mean value) of $x$.

Example: Gauss distribution

$$\rho(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-x_0)^2}{2\sigma^2}} \leadsto \langle x \rangle =\int_{-\infty}^{\infty}dx \rho(x)x=x_0.$$ (33)

If $f(x)$ is a function of the random variable $x$ (example: $x$ is the value of a position measurement, $f(x)=V(x)$ is the value of an external, fixed electric potential at $x$; $x$ random $\leadsto f(x)$ random as well), we define the

Definition 5 (Expectation value of a function)   : The expectation value of a function $f(x)$ is defined as

$$\fbox{$ \displaystyle \langle f \rangle :=\int_{-\infty}^{\infty}dx \rho(x)f(x). $}$$ (34)

Note that (1.33) is a special case of (1.34) with $f(x)=x$. An important special case of (1.33) is the

Definition 6 (Mean-square deviation)   The mean-square deviation of $x$ is defined as

$$\fbox{$ \displaystyle \langle \Delta x^2 \rangle \equiv \langle \left(x-\langle x\rangle \right)^2 \rangle. $}$$ (35)

Its value indicated how broadly scattered the individual realisations of x around its mean value $\bar{x}$ are. Example: Gauss distribution

$$\rho(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{x^2}{2\sigma^2}} \leadsto \langle \Delta x^2 \rangle =\int_{-\infty}^{\infty}dx \rho(x)x^2=\sigma^2.$$ (36)

In this example, we have used the trick of differentiation with respect to a parameter in order to calculate the integral

$$\int_{-\infty}^{\infty}dx x^2 e^{-a x^2} =-\frac{\partial}{\parti... ...c{\partial}{\partial a} \sqrt{\pi}a^{-1/2} = \frac{1}{2}\sqrt{\frac{\pi}{a^3}}.$$ (37)