Probability standard normal distribution

From the Standard Normal Distribution (SND) it is possible to determine the probability of negative elearanee, P. 

The Burchell model s prediction of the tensile failure probability distribution for grade H-451 graphite, from the "SIFTING" code, is shown in Fig. 23. The predicted distribution (elosed cireles in Fig. 23) is a good representation of the experimental distribution (open cireles in Fig. 23)[19], especially at the mean strength (50% failure probability). Moreover, the predicted standard deviation of 1.1 MPa con ares favorably with the experimental distribution standard deviation of 1.6 MPa, indicating the predicted normal distribution has approximately the correct shape. 

Mathematica hasthisfunctionandmanyothersbuiltintoitssetof "add-on" packagesthatare standardwiththesoftware.Tousethemweloadthepackage "Statistics NormalDistribution The syntax for these functions is straightforward we specify the mean and the standard deviation in the normal distribution, and then we use this in the probability distribution function (PDF) along with the variable to be so distributed. The rest of the code is self-evident. 

If T is normally distributed witli mean p and standard deviation a, then tlie random variable (T - p)/a is normally distributed with mean 0 and standard deviation 1. The term (T - p)/a is called a standard normal variable, and tlie graph of its pdf is called a "standard normal curve. Table 20.5.2 is a tabulation of areas under a standard normal cur e to tlie right of Zo of r normegative values of Zo. Probabilities about a standard normal variable Z can be detennined from tlie table. For example,... 

Table 20.5.2 also can be used to determine probabilities concerning normal random variables tliat are not standard normal variables. The required probability is first converted to tm equivalent probability about a standard normal variable. For example if T, the time to failure, is normally distributed with mean p = 100 and stanchird deviation a = 2 tlien (T - 100)/2 is a standard normal variable and... 

The probability function for the standard normal distribution is then... 

For np > 5 and n( 1 - p) > 5, an approximation of binomial probabilities is given by the standard normal distribution where z is a standard normal deviate and... 

The standard way to answer the above question would be to compute the probability distribution of the parameter and, from it, to compute, for example, the 95% confidence region on the parameter estimate obtained. We would, in other words, find a set of values h such that the probability that we are correct in asserting that the true value 0 of the parameter lies in 7e is 95%. If we assumed that the parameter estimates are at least approximately normally distributed around the true parameter value (which is asymptotically true in the case of least squares under some mild regularity assumptions), then it would be sufficient to know the parameter dispersion (variance-covariance matrix) in order to be able to compute approximate ellipsoidal confidence regions. 

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. 

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. 

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 explains graphically how probabilities and quantiles are defined for a normal distribution. For instance the 1 %-percentile (p = 0.01) of the standard normal distribution is —2.326, and the 99%-percentile (p 0.99) is 2.326 both together define a 98% interval. 

FIGURE 2.4 Probability density function of the uniform distribution (left), and the logit-transformed values as solid line and the standard normal distribution as dashed line (right). 

The probability that the variable x takes a value between a and b is given by the area under the graph of the probability distribution between x=a and x=b. This is illustrated in Figure 21.3, where the shaded area gives the probability that the standard normal variate, Z, lies between z and infinity, i.e. the probability P(Z>z). The total area under the graph is equal to 1, and because of the symmetry of the normal distribution it follows that the area of any one half is equal to 0.5. For any normal... 

Figure 21.3 The standard normal distribution curve. The shaded area gives the probability that the standard normal variate, Z, lies between z and infinity, i.e. P(Z>z).

Figure 2.1. The standardized normal or Gaussian distribution. The shaded area as a fraction as the entire area under the curve is the probability of a result between Xj and X2.

If jc has a normal distribution with mean p and standard deviation a, what is the probability distribution of... 

Using Table 1.3 i.e. Table B for standard normal distribution, determine probabilities that correspond to the following Z intervals. 

The chi-square distribution was discussed briefly in the earlier section on probability distributions. Suppose we have (k+1) independent standard normal variables. We then define % as the sum of the squares of these (k+1) variables. It can be shown that the probability density function of % is ... 

The control limits in Fig. 8-46 (UCL and LCL) are based on the assumption that the measurements follow a normal distribution. Figure 8-47 shows the probability distribution for a normally distributed random variable x with mean LI and standard deviation a. There is a very high probability (99.7 percent) that any measurement is within 3 standard deviations of the mean. Consequently, the control limits for x are typically chosen to be T 3, where a is an estimate of O. This estimate is usually determined from a set of representative data for a period of time when the process operation is believed to be typical. For the common situation in which the plotted variable is the sample mean, its standard deviation is estimated. 

When constructing input distributions for an uncertainty analysis, it is often useful to present the range of values in terms of a standard probability distribution. It is important that the selected distribution be matched to the range and moments of any available data. In some cases, it is appropriate to simply use the raw data or a custom distribution. Other more commonly used standard probability distributions include the normal distribution, the lognormal distribution, the uniform distribution, the log-uniform distribution and the triangular distribution. For the case-study presented below, we use lognormal distributions. 

According to the important theorem known as the central limit theorem, if N samples of size n are obtained from a population with mean, fi, and standard deviation, a, the probability distribution for the means will approach the normal probability distribution as N becomes large even if the underlying distribution is nonnormal. For example, as more samples are selected from a bin of pharmaceutical granules, the distribution of N means, x, will tend toward a normal distribution with mean /j and standard deviation <7- = a/s/n, regardless of the underlying distribution. 

In order to compensate for the uncertainty incurred by taking small samples of size n, the / probability distribution shown in Figure 3.2 is used in the calculation of confidence intervals, replacing the normal probability distribution based on z values shown in Figure 3.1. When n > 30, the /-distribution approaches the standard normal probability distribution. For small samples of size n, the confidence interval of the mean is inflated and can be estimated using Equation 3.9... 

Figure 19. The most probable C2-symmetric shape for a set of measurements after varying the probability distribution of the bottom measurement. Distributions are normal distributions marked as rectangles having width and length proportional to the standard deviation.

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