Introduction

There is a growing interest in probability distributions related to random variables which may spread over many orders of magnitudes or, within the same experimental discipline, there are abundant cases in which the spread is modest (less than one order of magnitude) or even undetectable. An example is Nuclear Magnetic Relaxometry, where one encounters systems with spreads of 3-4 orders of magnitude in the distribution of longitudinal magnetization decay rates (T₁ relaxation times), as well as systems with nearly mono-exponential decays.

Recently, this has led to some discussion [21] and to a considerable confusion when results of Inverse Laplace Transform [9] algorithms [10-20] of such relaxation decays started to be presented in terms of probability densities for different random variables (T₁ and log(T₁)) linked to the same probability distribution which, in addition, are plotted in graphs with different types of horizontal axes.

The purpose of this Note is to review the pertinent sections of probability theory, analyze the possible sources of confusion, illustrate the pertinent aspects with practical examples, and show that much of the controversy is a false problem which has nothing to do with the Laplace transform inversion algorithms.

Theory

Let Ω = {R,B₁,P(r)} be the usual probability space [1-6] in the set of real numbers R with its conventional Borel field (sigma-algebra) of open intervals B₁{R}.

For the sake of simplicity, we assume that the corresponding probability distribution function

is everywhere continuous.

Now let ρ_x and ρ_z be two probability density functions (pdf) in Ω, corresponding to two different, real-valued random variables x and z such that

where f(x) is a Borel function (one which has an inverse and which maps any random variable into a legitimate random variable). As usual, a pdf ρ(r) for a random variable r is defined indirectly by means of the integral

For continuous D(r), the existence of a ρ(r) satisfying Eq.(3) is guaranteed by one of the basic theorems of probability theory [5,7]. In this very particular case it is also possible to use simple Riemann integrals rather than the Lebeque-Stieltjes integrals [8] which, in the field of probability theory, are otherwise mandatory.

In the following, we are assuming that the two pdf's ρ_x and ρ_z both correspond to the same probability distribution D(r) specified by Eq.(1). Consequently, the change of random variable from x to z implies a corresponding change in the shape of the probability density function. In general, in fact, ρ_x(x) ≠ ρ_z(f(x)).

To derive the formulae for transformation between ρ_x and ρ_z, consider first that, by Eq.(3),

which, in differential form, gives

Replacing x by f^-1(z), one obtains

where f'(x) is the derivative of f(x). Consequently,

Eq.(7) provides the practical recipe for computing the pdf ρ_z when ρ_x is known.

Note added on March 30, 2006: 
   Some readers appear to be confused by the notation used for the derivative and inverse functions in the above equations. The following comments might help to clarify the situation (in order to enhance the arguments, we use the letter t for the independent variable wherever the choice of the symbol is actually irrelevant):
* f ^-1(t) denotes the function (mapping) which is the inverse of f(t). It may not be confused with 1/f(t) which is a different thing.
By definition, f ^-1(f(t)) = f(f ^-1(t)) = t. As an example, consider f(t) = t². Then f ^-1(t) = t^1/2.
* From Eq.(2) it follows that x = f ^-1(z), the value to be substituted into Eq.(5). For the present example x = z^1/2.
* The differential df ^-1(z) in the second section of Eq.(6) of course equals the derivative of f ^-1(z) with respect to z, multiplied by dz.
* The derivative of an inverse function is given by the textbook identity d[f ^-1(t)]/dt = 1/f '(f ^-1(t)), where f ' is the derivative of f,
i.e., f '(t) = df(t)/dt. This justifies the last passage in Eq.(6). In our example, d[f ^-1(z)]/dz = dz^1/2/dz = 1/(2z^1/2).
Since f '(t) = 2t, we have also 1/f '(f ^-1(z)) = 1/(2f ^-1(z)) = 1/(2z^1/2) and, as expected, the two expressions are identical.
* We have thus correctly established that when ρ_x(x) is the p.d.f. for x and z = x², then the p.d.f. for z is ρ_z(z) = ρ_x(z^1/2)/(2z^1/2).

Extensions

The above analysis can be extended to monotonously decreasing transformations functions f(x) such that -f(x) is a legitimate Borel function (e.g., f(x) = e^-x). In such a case, the differentials dx and dz in Eg.(5) are of opposite sign so that, in order to keep the pdf's positive, one needs to replace them by their absolute values. Consequently, Eq.(7) becomes

Another extension is to pdf's defined over sub-intervals (possibly different ones) of R. Thus, for example, when x is a random variable defined in [0,∞], z = ln(x) is a r.v. defined in [-∞,∞]. Such situations do not require any modification of Eq.(8) which handles them in an almost trivial way.

Examples

In the following examples we use the term scale to indicate the considered functional transform. As we shall see later, this may be a bit misleading since it does not necessarily match the horizontal-axis scale on which the probability density function is plotted. On the other hand, there are good reasons to consider the functional scale as the most natural (native) choice of the horizontal-axis scale in plots (which is why the term is admissible at all in the context of this Section).

A) Conversion from linear scale to natural logarithmic scale, z = ln(x).

In this case, f(x)= ln(x), f^-1(z)= e^z, f'(x)= 1/x, f'(f^-1(z))= 1/e^z and Eq.(8) gives

ρ_ln(z) = e^z.ρ_lin(e^z).

B) Conversion from linear scale to a logarithmic one in base 10 (Napier's), z= log(x).

Now f(x)= log(x), f^-1(z)= 10^z, f'(x)= log(e)/x and f'(f^-1(z))= log(e)/10^z. Hence

ρ_log(z) = 10^z.ρ_lin(10^z) /log(e).

C) Conversion from Napier scale to linear scale, z= 10^x.

In this case, f(x)= 10^x, f^-1(z)= log(z), f'(x)= 10^x ln(10)= 10^x /log(e) and f'(f^-1(z))= z/log(e). Hence

ρ_lin(z) = log(e).ρ_log(log(z)) / z.

D) Conversion from linear scale to inverse scale, z= 1/x.

We have f(x)= 1/x, f^-1(z)= 1/z, f'(x)= -1/x² and f'(f^-1(z))= z². Consequently

ρ_inv(z) = ρ_lin(1/z)/z².

E) Conversion from Napier scale to inverse scale, z= 1/10^x.

We have f(x)= 10^-x, f^-1(z)= -log(z), f'(x)= -10^x ln(10)= -10^x /log(e) and f'(f^-1(z))= -1/(z.log(e)). Hence

ρ_inv(z) = log(e).z.ρ_log(-log(z)).

The following Figure shows a sample distribution in terms of pdf's for the random variables X, log(X) and 1/X, each plotted using its native horizontal-axis scale.

Possible sources of confusion

Let us now limit our discussion to just two different random variables: X and log(X). We shall consider a single statistical distribution for X given by [1], which, through the elementary recipe

defines also the distribution of log(X).

While the distribution is the same, the corresponding probability density functions ρ_llin and ρ_log are different according to whether they are computed for the random variable X (linvar case) or for the random variable log(X) (logvar case). This is further complicated by the fact that both functions can be graphically displayed using either linear or logarithmic horizontal axis (linaxis or logaxis, respectively). The two things have nothing in common, all four cross-combinations are perfectly legitimate, as long as one knows exactly which is the chosen case. The possible confusion arises from the fact that the same term 'scale' is often applied to both the chosen random variable (X or log(X)) and the horizontal axis used to plot the functions (this latter aspect being just a mere display option which has nothing to do with probability theory).

In order to reduce the sources of confusion, however, it is recommendable to always represent linvar probability densities using graphs with linear axes and logvar probability densities using graphs with logarithmic axes. Only in these cases, in fact, can one apply the intuitive Riemann integration and avoid the less instinctive Lebesque integral with a non-linear measure.

As an example, Figure 2. shows a probability distribution represented by means of probability density functions for the variables X and log(X), each expressed using both linear and logarithmic horizontal axes. Though the four graphs appear different from each other, they are all legitimate and they all represent exactly the same statistical distribution.