Standard normal cumulative distribution function

Table A.l (Appendix 2) gives the proportion of values, F z), that lie below a given value of z. F z) is called the standard normal cumulative distribution function. For example the proportion of values below z = 2 is F(2) = 0.9772 and the proportion of values...

Table A.1 F[z), the standard normal cumulative distribution function...

The derivation of this formula is provided in Appendix 13B and Appendix 13C. Here, Fg is the standard normal cumulative distribution function and fg is the standard normal density function discussed in Appendix 12A in Chapter 12. The expected profit from ordering 0 units is evaluated in Excel using Equations 12.22,12.25, and 12.26, as follows ... 

Here, fix) is the normal density function, fs ) is the standard normal density function, and Fsi,) is the standard normal cumulative distribution function. 

Assume that the manufacturer incurs a production cost of v per unit and charges a wholesale price of c from the retailer. The retailer, in turn, sells to customers for a price of p. The retailer salvages any leftover units for sr. The manufacturer salvages any leftover units for %. If retailer demand is normally distributed, with a mean of /r and a standard deviation of a, we can evaluate the impact of a quantity flexibility contract. If the retailer orders 0 units, the manufacturer is committed to supplying Q units. As a result, we assume that the manufacturer produces Q units. The retailer purchases q units if demand D is less than q, D units if demand D is between q and Q, and Q units if demand D is greater than Q. In the following formulas, Fs is the standard normal cumulative distribution function and fs is the standard normal density function discussed in Appendix 12A in Chapter 12. We thus obtain... 

Transform the original Z data to a standard normal distribution (all work will be done in normal space). There are different techitiques for this transformation. The normal score transformation whereby the normal transform y is calculated from the original variable z as y = G- [F(z)], where G(-) is the standard normal cumulative distribution function (cdf) and F -) is the cdf of the original data. 

The normal probability function table given in the appendix d this book can also be used for values of the log-normal distribution function, f, and the log-normal cumulative distribution function, F. In these tables Z = [ln(d/cy/(In o- )] is used. A plot of the cumulative log-normal distribution is linear on log-normal probability paper, like that shown in Figure 2.11. A size distribution that fits the log-normal distribution equation can be represented by two numbers, the geometric mean size, dg, and the geometric standard deviation,. The geometric mean size is the size at 50% of the distribution, d. The geometric standard deviation is easily obtained finm the following ratios ... 

Therefore, it is common to assume that they are well modeled by log-normal cumulative distribution functions F (standard deviation a. The log-normal cumulative distribution function is defined as... 

Figure 4.6 Shape of the Cumulative Distribution Function (CDF) for an arbitrary normal distribution with varying standard deviation (adapted from Carter, 1986)...

The normal probability distribution function can be obtained in Microsoft Excel by using the NORMDIST function and supplying the desired mean and standard deviation. The cumulative value can also be determined. In MATLAB, the corresponding command is randn. 

The normal distribution, A Y/l, o 2), has a mean (expectation) fi and a standard deviation cr (variance tr2). Figure 1.8 (left) shows the probability density function of the normal distribution N(pb, tr2), and Figure 1.8 (right) the cumulative distribution function with the typical S-shape. A special case is the standard normal distribution, N(0, 1), with p =0 and standard deviation tr = 1. The normal distribution plays an important role in statistical testing. 

FIGURE 1.8 Probability density function (PDF) (left) and cumulative distribution function (right) of the normal distribution cr2) with mean /a and standard deviation cr. The quantile q defines a probability p. 

Parameter Two distinct definitions for parameter are used. In the first usage (preferred), parameter refers to the constants characterizing the probability density function or cumulative distribution function of a random variable. For example, if the random variable W is known to be normally distributed with mean p and standard deviation o, the constants p and o are called parameters. In the second usage, parameter can be a constant or an independent variable in a mathematical equation or model. For example, in the equation Z = X + 2Y, the independent variables (X, Y) and the constant (2) are all parameters. 

To compute the integral in (9), we use the fact that, for fixed a%2, a x, the quantity Pidki - 9k2 > (o 2 aki)/ )is ttie cumulative distribution function of a standard normal distribution evaluated at... 

The mean and standard deviation of the normal distribution are T and a, respectively. Since the normal distribution is designed for continuous data, the cumulative distribution function is more practical than the probability density function. For a particular data population, the cumulative distribution [2] is as follows ... 

Making this substitution into Equation (3.6) or (3.7) reduces the generic normal distribution to one with mean 0 and standard deviation 1, collapsing all possible normal distributions onto a standard curve. Tabulated values of the cumulative distribution function F are usually presented in terms of the transformation variable z. Sample values of F(z) are presented in Table 3.2. Microsoft Excel contains an intrinsic function, NORMSDIST, that produces the cumulative probability for a standard normal variable z given as its argument. A companion function, NORMSINV, outputs the z value for a given F(z). The Microsoft Excel manual or the electronic help files [5, 6] provide command syntax and usage examples. 

Thus, (z) is the cumulative distribution function and < z) is the probability density function, respectively, for the standard normal variable Z. The failure rate for the normal distribution is a mon-otonicaUy increasing function. Normal distribution should be used as a life distribution when ix > 6a because then the probabihty that T will be negative is exceedingly small. Otherwise, truncated normal distribution should be used. 

Where a random variable X is assumed to follow normal distribution it will be described in the fovmXa ) where means is distributed according to . Examples of the graphical representation of the normal distribution are included in Chapter 37. The standard normal distribution is written as N(0,1) with j = 0 and r7 = 0. The cumulative distribution function for the standard normal distribution is given by... 

The OC function is the cumulative distribution function of the standard normal distributed variable Z ... 

Recall that F. ) is the cumulative distribution function and/ C.) is the probability density function for the standard normal distribution with mean 0 and standard deviation 1. Using Equation 12.21 and the definition of the standard normal distribution, we have... 

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