Normal random number generator

An empirical test of this possibility was performed by computing values of the variance of the two terms in equation 44-77. The Normal random number generator of MATLAB was used to create multiple values of Normally distributed random numbers for Er and Es these were plugged into the two expressions of equation 44-77 and the variance computed. Values between 100 and 106 were used in each computation of the variance. 

To do the computations, we again use the random number generator of MATLAB to produce Normally-distributed random numbers with unity variance to represent the noise values of Er will then directly represent the S/N ratio of the data being evaluated. For the computations reported here, we use 100,000 synthetic values of the expression on the RHS of equation 44-76a to calculate the variance of, for each combination of conditions we investigate. 

Models are often best understood relative to the situation they are designed to describe if their constitutive variables are allowed to fluctuate statistically in a realistic way. Once a variable has been assigned a suitable density of probability distribution and the parameters of this distribution have been chosen, the fluctuations can be conveniently produced by using random deviates from statistical tables. A random deviate is a particular value of a standard random variable. Many elementary books in statistics contain tables of deviates from uniform, normal, exponential,. .. distributions. Many high-level computation-oriented programming languages (e.g., MatLab) and spreadsheets, such as Microsoft Excel, also contain random number generators. The book by Press et al. (1986) contains software that produces random deviates for the most commonly used probability distributions. 

A second facility that is sometimes useful is the random number generator function. There are several possible distributions, but the most usual is the normal distribution. It is necessary to specify a mean and standard deviation. If one wants to be able to return to the distribution later, also specify a seed, which must be an integer number. Figure A. 15 illustrates the generation of 10 random numbers coming from a distribution of mean 0 and standard deviation 2.5 placed in cells A1 -A10 (note that the standard deviation is of the parent population and will not be exactly the same for a sample). This facility is very helpful in simulations and can be employed to study die effect of noise on a dataset. 

Suppose that using Monte Carlo simulation witli 10 simulated values of Ta and 10 simulated values of Tq, it is desired to estimate an average value of Ts. First, 20 random numbers are generated. Tliese are shown in columns 1 and 4 of Table 20.6.2. Regard each of the random numbers generated as the value of tlie cdf of a standard normal variable Z. Let Zi be tlie simulated value of Z corresponding to 0.10, tlie first random number in colunm 1. Tlien, since 0.10 is tlie value of tlie cdf for Z = Zi,... 

In a similar way, a 50 50 two-way fluorescence EEM of a five-component system was generated. Components d and e were taken as the sought-for analytes with a relative concentration 0.5 and 1.0 respectively, and the other three were regarded as unknown interferents with the same relative concentration of 0.5. Zero-mean normal random numbers were added to the mixture EEM to simulate the experimental noise and the standard deviation of the noise was taken to be one-thousandth of the largest value in the mixture EEM. This data set is used for the comparison of GSA with the Powell algorithm. 

The most recent revision of MATLAB Version 4.0 for Windows, contains modifications in the random number generator options. In MATLAB 4.0, the rand command generates numbers from a uniform distribution between 0 and 1. The command randn is used for random generation of values within the unit normal distribution (N(0,1)). Previous versions used the rand command for both distributions with the switches rand( normar) and rand( uniform ) to designate the distribution to use on subsequent rand commands. While the... 

Monte Carlo simulation is a procedure for mimicking observations on a random variable that permits verification of results that would ordinarily require difficult mathematical calculations or extensive experimentation. The method normally uses computer programs called random number generators. A random number is a number selected from the interval (0,1) in such a way that the probabilities that the number comes from any two subintervals of equal length are equal. For example, the probability the number is in the subinterval (0.1, 0.3) is the same as the probability that the number is in the subinterval (0.5, 0.7). Thus, random numbers are observations on a random variable X having a uniform distribution on the interval (0,1). This means that the PDF of X is specified by... 

The random number generated is the cumulative probability, and the cumulative probability is the area under the standard normal distribution curve. Since the standard normal distribution curve is symmetrical, the negative values of Z and the corresponding area are found by symmetry. For example, as described in the two previous problems,... 

In the Random Number Generation dialogbox that now appears, select Normal as the Distribution, 10 as the Mean, and 1 as the Standard Deviation. Furthermore, specify the Output Range as B3 B302. Click OK. 

Fill cells B6 B15 with Gaussian noise with zero average (or mean ) and unit standard deviation. (Reminder such Gaussian noise can be found under Tools O Data Analysis => Random Number Generation O Distribution Normal, Mean = 0, Standard Deviation = 1, Output Range B6 B15 => OK.)... 

In column F, starting with cell F6, deposit Gaussian noise. (Select Tools Data Analysis, highlight Random Number Generation, click OK, select Distribution Normal, Mean = 0, Standard Deviation = 1, click on Output Range, then click on the adjacent window, specify the output range, and click OK.)... 

In column B deposit Gaussian noise (Tools O Data Analysis O Random Number Generation => Distribution Normal, Mean 0, Standard Deviation 1). 

In some out-of-viewplacelike N10 Q40, enter some Gaussian noise (using Tools Data Analysis... => Random Number Generation, OK4> Distribution Normal,... 

After completing the computation of the mean Ky° and KyC and standard deviations, a random number generator was used to develop 1,000 normally distributed values of each Kv° and Kv°, having the listed standard deviations. Verification of the normality of the randomly generated number sets was done by a Chi-square test on each number set. 

It is only the first class of applications to which this chapter is devoted, because these computations require the highest quality of random numbers. The ability to do a multidimensional integral relies on properties of uniformity of n-tuples of random numbers and/or the equivalent property that random numbers be uncorrelated. The quality aspect in the other uses is normally less important simply because the models are usually not all that precisely specified. The largest uncertainties are typically due more to approximations arising in the formulation of the model than those caused by lack of randomness in the random number generator. 

A normal distribution was assumed, with interlaboratory relative standard deviations (RSD) of 25% assumed at 50 and 100 (xg/kg, and 20% at 200 p.g/kg. These assumptions are based on the Horwitz equation. The nominal means and RSD assumptions were input into the random-number generator to obtain mean laboratory values. 

A normal distribution of the ratio is assumed, and that analyte concentration in the PT samples varies from 50 to 200 p-g/kg in the samples. At these concentrations, the Horwitz equation predicts interlaboratory relative standard deviations of 20-25%. An average ratio of 1, with inter-round relative standard deviations (RSDs) of 25%, are input into the random-number generator to obtain individual ratios for the analyte from each of the 10 PT rounds. 

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