vstatmp_engl

40 3 Probability Distributions and their Properties

Fig. 3.9. Transformation of a p.d.f. f(x) into g(u) with u = x2. The sum of the shaded areas below f(x) is equal to the shaded area below g(u).

Fig. 3.10. Transformation of a uniform distribution of the radius into the distribution of the volume of a sphere.

we ask for the distribution g(V ) of the volume V (r). With

Example 23. Distribution of the quadratic deviation

For a normal distributed variate x with mean value x0 and variance s2 we ask for the distribution g(u), where

u = (x − x0)2/s2

is the normalized quadratic deviation. The expected value of u is unity, since the expected value of (x − µ)2 per definition equals σ2 for any distribution. The function x(u) has two branches. With

The contributions from both branches are the same, thus

Example 24. Distribution of kinetic energy in the one-dimensional ideal gas Be v the velocity of a particle in x direction with probability density

Its kinetic energy is E = v2/(2m), for which we want to know the distribution g(E). The function v(E) has again two branches. We get, in complete analogy to the example above,

The contributions of both branches are the same, hence

g(E) = √ 1 e−E/kT . πkT E

3.4.2 Determination of the Transformation Relating two Distributions

In the computer simulation of stochastic processes we are frequently confronted with the problem that we have to transform the uniform distribution of a random number generator into a desired distribution, e.g. a normal or exponential distribution.

42 3 Probability Distributions and their Properties

More generally, we want to obtain for two given distributions f(x) and g(u) the transformation u(x) connecting them.

Integrating, we get F (x) and G(u):

F (x) = G(u) ,

u(x) = G−1 (F (x)) .

G−1 is the inverse function of G. The problem can be solved analytically, only if f and g can be integrated analytically and if the inverse function of G can be derived.

Let us consider now the special case mentioned above, where the primary distribution f(x) is uniform, f(x) = 1 for 0 ≤ x ≤ 1. This implies F (x) = x and

Example 25. Generation of an exponential distribution starting from a uniform distribution

The desired transformation u(x), as demonstrated above in the general case, is obtained by integration and inversion:

Z u Z x

g(u′) du′ = f(x′) dx′ ,

Z u0 Z0x

λe−λu′ du′ = f(x′) dx′ ,

0 0

1 − e−λu = x ,

u = − ln(1 − x)/λ .

We could have used, of course, also the relation (3.32) directly. Obviously in the last relation we could substitute 1 − x by x, since both quantities are uniformly distributed.

When we transform the uniformly distributed random numbers x delivered by our computer according to the last relation into the variable u, the latter will be exponentially distributed. This is the usual way to simulate the lifetime distribution of instable particles and other decay processes (see Chap. 5).

3.5 Multivariate Probability Densities

The results of the last sections are easily extended to multivariate distributions. We restrict ourself here to the case of continuous distributions8.

8An example of a multivariate discrete distribution will be presented in Sect. 3.6.2.

3.5.1 Probability Density of two Variables

Definitions

As in the one-dimensional case we define an integral distribution function F (x, y), now taken to be the probability to find values of the variates x′, y′ smaller than x, respectively y

The following properties of this distribution function are satisfied:

F (∞, ∞) = 1,

F (−∞, y) = F (x, −∞) = 0 .

In addition, F has to be a monotone increasing function of both variables. We define a two-dimensional probability density, the so-called joined probability density, as the partial derivation of f with respect to the variables x, y:

From these definitions follows the normalization condition

Z ∞ Z ∞

f(x, y) dx dy = 1 .

−∞ −∞

The projections fx(x) respectively fy(y) of the joined probability density onto the coordinate axes are called marginal distributions :

Z ∞

fx(x) = f(x, y) dy ,

Z−∞∞

fy(y) = f(x, y) dx .

−∞

The marginal distributions are one-dimensional (univariate) probability densities.

The conditional probability densities for fixed values of the second variate and normalized with respect to the first one are denoted by fx(x|y) and fy(y|x) for given values of y or x, respectively. We have the following relations:

44 3 Probability Distributions and their Properties

Example 26. Superposition of two two-dimensional normal distributions9 The marginal distributions fx(x), fy(y) and the conditional p.d.f.

fy(y|1) and fR(y, 1) di er in the normalization factor, which results from the requirement fy(y|1) dy = 1.

Graphical Presentation

Fig. 3.11 shows a similar superposition of two Gaussians together with its marginal distributions and one conditional distribution. The chosen form of the graphical representation as a contour plot for two-dimensional distributions is usually to be favored over three-dimensional surface plots.

3.5.2 Moments

Analogously to the one-dimensional case we define moments of two-dimensional distributions:

Explicitly,

9The two-dimensional normal distribution will be discussed in Sect. 3.6.5.

Fig. 3.11. Two-dimensional probability density. The lower left-hand plot shows the conditional p.d.f. of y for x = 2. The lower curve is the p.d.f. f(y, 2). It corresponds to the dashed line in the upper plot. The right-hand side displays the marginal distributions.

Z Z

µ′lm = (x − µx)l(y − µy)mf(x, y) dx dy .

−∞ −∞

Obviously, µ′x, µ′y (= µ′10, µ′01) are zero.

Correlations, Covariance, Independence

The mixed moment σxy is called covariance of x and y, and sometimes also denoted as cov(x, y). If σxy is di erent from zero, the variables x and y are said to be correlated. The mean value of y depends on the value chosen for x and vice versa. Thus, for instance, the weight of a man is positively correlated with its height.

The degree of correlation is quantified by the dimensionless quantity

ρxy = σxy ,

σxσy