Many phenomena in the real world are described very well by Gaussian or "normal" distribution functions.

ListPlot[{{290, 0}, {300, 0.02}, {310, 0}, {320, 0.04}, {330, 
            0.06}, {340, 0.11}, {350, 0.14}, {360, 0.16}, {370, 0.15}, {380, 
            0.08}, {390, 0.10}, {400, 0.08}, {410, 0.02}, {420, 0.03}, {430, 
            0}, {440, 0.01}, {450, 0}}, 
        PlotJoined -> True];

This function has a simple basic form:

Plot[Exp[-x^2], {x, -3, 3}];

A more complicated (and more familiar?) form is the following. Note that it is identical to Eq. 13-7 in Levenspiel, with (σ^2)=2(D/uL) (see Figure 13.4).

f(x) = (1/(Sqrt[2  π(σ^2)])) 
            (e^(-((((x - μ))^2)/(2  (σ^2)))))

sigma = 1;  
mu = 3;  
gaussian = (1/(2  Pi*sigma^2)^0.5)*
        Exp[-(x - mu)^2/(2  sigma^2)]
Plot[gaussian, {x, 0, 6}, 
        PlotRange -> {0, 0.5}];
Integrate[gaussian, {x, 2, 4}]

One of the most important concepts in PChem is the Maxwell-Boltzmann distribution of velocities:

f((Subscript[v, x]))=((((M/(2  πRT))))^0.5)(e^(-(((Subscript[Mv, x]^2)/(2  RT)))))

molWeight = 0.032;  
gasConst = 
        8.314;   
temp = 773;

velDistr = (molWeight/(2*Pi*gasConst*temp))^0.5*
        Exp[-molWeight*
            vX^2/(2  gasConst*temp)]
Plot[velDistr, {vX, -1200, 1200}];
Integrate[
      velDistr, {vX, -Infinity, +Infinity}]
meanVel = (8  gasConst*temp/Pi/molWeight)^0.5

These are all symmetric curves, described by two parameters, mean and variance. The most famous skewed distribution function is the log-normal one. Perhaps an even greater number of real-world phenomena conform to this distribution function:

f(x) = (1/(x (Sqrt[ 2  π(σ^2)]))) 
            (e^(-((((ln(x) - μ))^2)/(2  (σ^2)))))

sigma = 2;  
mu = 3;  
logNormal = (1/(x*(2  Pi*sigma^2)^0.5))*
        Exp[-(Log[x] - mu)^2/(2 
                 sigma^2)]
Plot[logNormal, {x, 0, 10}, 
        PlotRange -> {0, 0.1}];

LRR3@psu.edu