Beyond the mean and the variance

The mean and the standard deviation are the two most important parameters of a PD . We note here some others which you are likely to meet occasionally.

• The mode of a PD locates its maximum and thus identifies the most probable value of the associated variable.

• The k-th moment of a PD, $\mu_k$, is defined by:

  $$\mu_k=\langle x^k \rangle$$

  Thus in particular the first moment, $\mu_1$, is simply the mean:

  $$\mu_1 = \langle x \rangle$$

  while the first and second moments together prescribe the variance (see KP3.9 )

  $$V[x] = \langle (\Delta x)^2 \rangle= \langle x^2 \rangle -\langle x \rangle^2 =\mu_2 -\mu_1 ^2$$

  In some important cases the values of the first two moments (equivalently the mean and the variance) alone are sufficient to specify the PD. In other cases higher moments provide further details of the PD.

• The root-mean-square value of a random-variable is the square root of its second moment:

  $$x_{rms} \equiv \sqrt{\mu_2} = \sqrt{\langle x^2 \rangle}$$

By way of an example here is a first look at one of the most celebrated probability distributions in all of physics: it is known as the Maxwell distribution. Strictly, it is the probability density f(v) for the speed v of a moleculule in a gas, at temperature T.

Before too long (you will need some statistical physics) you will be able to show that it has the form:

f(v) = Av² e^-Bv2

where A and B are independent of v (but known functions of the molecular mass and the temperature).

At this point, however, you might just look at its form, displayed above.

There are three measures of a `typical' speed:

• the mean, or average speed: v_av

• the mode, or most probable speed: v_mp

• the root-mean-square speed: v_rms

They are marked. Note that they are different.

Establishing their values, from the pdf given above, and working out some of the other moments of this pdf will exercise your capacity to integrate by parts. Look forward (with pleasure) to TQ5 .!