vstatmp_engl

Fig. 3.14. Distribution of the di erence t between two times t1 and t2 which have both clock readings equal to zero.

h(t1, t) = f(t1, t2) = 1/Δ2 ,

and has the boundaries shown in Fig. 3.14. The form of the marginal distribution is found by integration over t1, or directly by reading it o from the figure:

where T = T1 − T2 now for arbitrary values of T1 and T2.

Example 31. Distribution of the transverse momentum squared of particle tracks

The projections of the momenta are assumed to be independently normally distributed,

50 3 Probability Distributions and their Properties

The result is an exponential distribution in p2 with mean p2 = p2x + p2y .

Example 32. Quotient of two normally distributed variates

For variates x, y, independently and identically normally distributed, i.e.

we want to find the distribution g(u) of the quotient u = y/x. Again, we transform first into new variates u = y/x , v = x, or, inverted, x = v , y = uv and get

h(u, v) = f(x(u, v), y(u, v)) ∂(x, y) , ∂(u, v)

where the substitution z = v2/2 has been used. The result g(u) is the Cauchy distribution (see Sect. 3.6.9). Its long tails are caused here by the finite probability of arbitrary small values in the denominator. This e ect is quite important in experimental situations when we estimate the uncertainty of quantities which are the quotients of normally distributed variates in cases, where the p.d.f. in the denominator is not negligible at the value zero.

The few examples given above should not lead to the impression that transformations of variates always yield more or less simple analytical expressions for the resulting distributions. This is not the rule, but rather the exception. However, as we will learn in Chap. 5, a simple, straight forward numerical solution is provided by Monte Carlo methods.

3.5.5 Determination of the Transformation between two Distributions

As in the one-dimensional case, for the purpose of simulation, we frequently need to generate some required distribution from the uniformly distributed random numbers delivered by the computer. The general method of integration and inversion of the cumulative distribution can be used directly, only if we deal with independent variates. Often, a transformation of the variates is helpful. We consider here a special example, which we need later in Chap. 5.

Example 33. Generation of a two-dimensional normal distribution starting from uniform distributions

We use the result from example 29 and start with the representation of the two-dimensional Gaussian in polar coordinates

g(ρ, ϕ) dρ dϕ = 21π dϕ ρ e−ρ2/2 dρ ,

which factorizes in ϕ and ρ. With two in the interval [0, 1] uniformly distributed variates r1, r2, we obtain the function ρ(r1):

In the same way we get ϕ(r2):

ϕ = 2πr2 .

These variables are independent and distributed normally about the origin with variance unity:

f(x, y) = 21π e−(x2+y2)/2 .

(We could replace 1 − r1 by r1, since 1 − r1 is uniformly distributed as well.)

3.5.6 Distributions of more than two Variables

It is not di cult to generalize the relations just derived for two variables to multivariate distributions, of N variables. We define the distribution function F (x1, . . . , xN ) as the probability to find values of the variates smaller than x1, . . . , xN ,

F (x1, . . . , xN ) = P {(x′1 < x) ∩ · · · ∩ (x′N < xN )} ,

and the p.d.f.

∂N F

f(x1, . . . , xN ) = ∂x1, . . . , ∂xN .

Often it is convenient to use the vector notation, F (x), f(x) with

x = {x1, x2, . . . , xN } .

These variate vectors can be represented as points in an N-dimensional space.

The p.d.f. f(x) can also be defined directly, without reference to the distribution function F (x), as the density of points at the location x, by setting

3.5.8 Angular Distributions

In physics applications we are often interested in spatial distributions. Fortunately our problems often exhibit certain symmetries which facilitate the description of the phenomena. Depending on the kind of symmetry of the physical process or the detector, we choose appropriate coordinates, spherical, cylindrical or polar. These coordinates are especially well suited to describe processes where radiation is emitted by a local source or where the detector has a spherical or cylindrical symmetry. Then the distance, i.e. the radius vector, is not the most interesting parameter and we often describe the process solely by angular distributions. In other situations, only directions enter, for example in particle scattering, when we investigate the polarization of light crossing an optically active medium, or of a particle decaying in flight into a pair of secondaries where the orientation of the normal of the decay plane contains relevant information. Similarly, distributions of physical parameters on the surface of the earth are expressed as functions of the angular coordinates.

Distribution of the Polar Angle

As already explained above, the expressions

x = r cos ϕ , y = r sin ϕ

relate the polar coordinates r, ϕ to the cartesian coordinates x , y. Since we have periodic functions, we restrict the angle ϕ to the interval [−π, π]. This choice is arbitrary to a certain extent.

For an isotropic distribution all angles are equally likely and we obtain the uniform distribution of ϕ

g(ϕ) = 21π .

Since we have to deal with periodic functions, we have to be careful when we compute moments and in general expected values. For example the mean of the two

angles ϕ1 = π/2, ϕ2 = −π is not (ϕ1 + ϕ2)/2 = −π/4, but 3π/4. To avoid this kind of mistake it is advisable to go back to the unit vectors {xi, yi} = {cos ϕi, sin ϕi}, to

average those and to extract the resulting angle.

Example 34. The v. Mises distribution

We consider the Brownian motion of a particle on the surface of a liquid. Starting from a point r0 its position r after some time will be given by the expression

Taking into account the Jacobian ∂(x, y)/∂(r, ϕ) = r, the distribution in polar coordinates is:

For convenience we have chosen the origin of ϕ such that ϕ0 = 0. For fixed r we obtain the conditional distribution

54 3 Probability Distributions and their Properties

g˜(ϕ) = g(ϕ|r) = cN (κ) exp (κ cos ϕ)

with κ = rr0/σ2 and cN (κ) the normalization constant. This is the v. Mises distribution. It is symmetric in ϕ, unimodal with its maximum at ϕ = 0. The

normalization

1 cN (κ) = 2πI0(κ)

contains I0, the modified Bessel function of order zero [22].

For large values of κ the distribution approaches a Gaussian with variance 1/κ. To demonstrate this feature, we rewrite the distribution in a slightly modified way,

g˜(ϕ) = cN (κ)eκe[−κ(1−cos ϕ)] ,

In the limit κ = 0, which is the case for r0 = 0 or σ → ∞, the distribution becomes uniform, as it should.

Distribution of Spherical Angles

Spatial directions are described by the polar angle θ and the azimuthal angle ϕ which we define through the transformation relations from the cartesian coordinates:

The Jacobian is ∂(x, y, z)/∂(r, θ, ϕ) = r2 sin θ. A uniform distribution inside a sphere of radius R in cartesian coordinates

thus transforms into

3r2

hu(r, θ, ϕ) = 4πR3 sin θ if r ≤ R .

We obtain the isotropic angular distribution by marginalizing or conditioning on r:

Spatial distributions are usually expressed in the coordinates z˜ = cos θ and ϕ, because then the uniform distribution simplifies further to

1 gu(˜z, ϕ) = 4π

with |z˜| ≤ 1.

The p.d.f. g(˜z, ϕ) of an arbitrary distribution of z,˜ ϕ is defined in the standard way through the probability d2P = g(˜z, ϕ)d˜zdϕ. The product d˜zdϕ = sin θdθdϕ = d2Ω is called solid angle element and corresponds to an infinitesimal area at the surface of the unit sphere. A solid angle Ω defines a certain area at this surface and contains all directions pointing into this area.

Example 35. Fisher’s spherical distribution

Instead of the uniform distribution considered in the previous example we now investigate the angular distribution generated by a three-dimensional rotationally symmetric Gaussian distribution with variances σ2 = σx2 = σy2 = σz2. We put the center of the Gaussian at the z-axis, r0 = {0, 0, 1}. In spherical coordinates we then obtain the p.d.f.

For fixed distance r we obtain a function of θ and ϕ only which for our choice of r0 is also independent of ϕ:

g(θ, ϕ) = cN (κ) sin θ exp(κ cos θ) .

The parameter κ is again given by κ = rr0/σ2. Applying the normalization

a two-dimensional, unimodal distribution, known as Fisher’s spherical distribution. As in the previous example we get in the limit κ → 0 the uniform distribution (3.41) and for large κ the asymptotic distribution

g˜(θ, ϕ) ≈ 41π κθ e−κθ2/2 ,

which is an exponential distribution of θ2. As a function of z˜ = cos θ the distribution (3.42) simplifies to

which illustrates the spatial shape of the distribution much better than (3.42).

3.6 Some Important Distributions

3.6.1 The Binomial Distribution

What is the probability to get with ten dice just two times a six ? The answer is given by the binomial distribution: