Solution to random error propagation challenge

The problem of finding the absorbance value that results in the lowest relative uncertainty of the analyte concentration is discussed. This problem can be solved with a Monte Carlo numerical approach which uses method in which the random errors (noise) in the light intensities I and I0 are simulated in the form of added random numbers, ε1 and ε2. The random numbers ε1 and ε2 have to be generated so that they each form a normal distribution with a mean of zero and a standard deviation of one can be achieved by using the Box-Muller transformation or by summing a dozen uniformly distributed random numbers. Monte Carlo simulation consists of setting I 0=1, σI0= σI=0.01 and varying the values of I from 0.30 to 0.40. The lowest relative uncertainty of the absorbance occurs at approximately equal to 0.48, and not at 0.434. The optimum absorbance value is determined analytically by differentiating with respect to I 0 and I, treating both as independent variables.