11.5 Moments and associated  11. Special topics  11.3 Nomograms

11.4 Transforming distribution functions

A differential probability distribution function f(x), to describe the probability associated with events having (possibly multidimensional) coordinate x, must have these properties:

• It must be normalized so that the integral over the defined space gives unity:

$$\int_{-\infty}^{\infty}f(x)dx = 1 \ .$$ (11.11)

 
• It must be positive definite:

$$f(x) \ge 0 \quad{\rm for~all~}x.$$ (11.12)

 
• The integral between specified limits must give the probability that events will occur in that interval:

$$\int_a^b f(x) ds = P(a<x<b) \ .$$ (11.13)

 

One often needs to transform such distribution functions to find probability distribution functions in terms of new variables. Consider the transformation from variable x to variable y where the relationship between y and x is known (y(x)). Then, if the probability distribution function g(y) represents the same probability distribution function in terms of the variable y, to preserve the relationship of integrals to probability it must be true that

$$\int_{x_1}^{x_2} f(x)dx = \int_{y(x_1)}^{y(x_2)} g(y)dy \ \ .$$ (11.14)

 

Differentiating gives:

$${rl}{{d}\over{dx_2}} \left[\int_{x_1}^{x_2} f(x)dx \right] = f(x_2)= {{d}\over{dx_2}} \left[\int_{y(x_1)}^{y(x_2)} g(y)dy\right]$$

$$={{d}\over{dy_2}} \left[\int_{y(x_1)}^{y(x_2)} g(y) dy\right]{{dy_2}\over{dx}}\vert _{x_2}$$

$$= g(y_2) {{dy_2(x_2)}\over{dx_2}} = g(y) {{dy}\over{dx}}\vert _{x_2} \ .$$ (11.15)

 

A shorthand way of representing this relationship is

$$f(x)dx = g(y)dy \ \$$ (11.16)

 

where dx and dy are understood to be corresponding infinitesimal intervals.^11.3

The transformation between variables is then, from (11.16),

$$f(x) = g(y){{dy}\over{dx}} \ \ .$$ (11.17)

 
 

Example 11.2: From a droplet distribution function in terms of radius, n(r), find the distribution functions f(x) in terms of droplet mass x and $g(\log_{10}(r))$ in terms of the base-10 logarithm of the radius.

Because

$$x={{4}\over{3}} \pi r^3 \rho_w,$$ (11.18)

 

$$dx = 4\pi r^2\rho_w dr$$ (11.19)

 

so

$$f(x) = {{n(r)}\over{4\pi r^2\rho_w}} \ .$$ (11.20)

 

The second part of the problem might be considered poorly stated. As explained in section 11.2, for dimensional consistency it might appear that the distribution function should be presented in terms of a dimensionless argument. The appropriate distribution function would then have the form $g(\log_{10}(r/r_0)$ where r₀ is a constant reference radius. Then

$${rl}{{d(\log_{10}(r/r_0))}\over{dr}} ={{d(\log_{10}(r/r_0))}\over{d(\ln_e(r/r_0))}}\times {{d(\ln_e(r/r_0))}\over{dr}}$$

$$= {{1}\over{\ln_e(10)}} {{1}\over{r}}$$ (11.21)

 

so

$$g(\log_{10}(r/r_0)) = \ln_e(10) r n(r) \approx 2.30 r n(r) \ \ .$$ (11.22)

 

Because the resulting distribution function does not depend on the value selected for r₀, the reference dimension r₀ is almost universally omitted from logarithmic distribution functions like this. There is no dimensional inconsistency in doing so because the logarithm is only offset by a fixed amount depending on the choice of units and hence the change in the logarithm remains the same for a given interval regardless of the choice of units. For example, if r is expressed in $\mu$m and the units of n(r) are cm^-3 $\mu$m^-1, the units of g(log₁₀(r/r₀)) are cm^-3 per logarithmic interval in radius, and this is the same for any other choice of unit for r.

A cumulative distribution function can be defined as the integral of the differential distribution function, so that

$$N(r) = \int_{-\infty}^r n(r^\prime)dr^\prime$$ (11.23)

 

is the number of events with radius (for example) smaller than r. Then the derivative of this cumulative distribution function with respect to any variable is the differential probability distribution function for that variable. For example,

$${{dN(\log_{10}(r))}\over{d\log_{10}(r)}} = g(\log_{10}(r)) \ .$$ (11.24)

 

For clarity, plots of differential distribution functions are often labeled with the form on the left side, rather than the right side, of this equation.

It is sometimes useful to display moments of a distribution function. For example, if x is the droplet mass as before, the distribution of mass in a droplet size distribution can be displayed as

$$h(x) = x f(x) = {{4}\over{3}} \pi r^3\rho_w{{n(r)}\over{r\pir^2\rho_w}}={{rn(r)}\over 3}$$ (11.25)

 

to produce a plot for which area is proportional to mass.
 

Exercise 11.1: Show that, if a logarithmic scale is needed, the appropriate function to show the distribution ( x dN/d log₁₀(r)) in mass is (4/3) $\pi r^4\ln_e(10)\rho_wn(r)$.

   
 11.5 Moments and associated  11. Special topics  11.3 Nomograms