The Standard Normal distribution

The evaluation of integrals involving the Gaussian distribution, such as those of Eqs. 2.56, 2.61, and 2.62, requires tedious numerical integration. The result of such integrations is a function of m and a-. Therefore, the calculation should be repeated every time w or o- changes. To avoid this repetition, the normal distribution is rewritten in such as way that 

The resulting function is called the standard normal distribution. Integrals involving the Gaussian distribution, such as that of Eq. 2.61, have been tabulated based on the standard normal distribution for a wide range of x values. With the help of a simple transformation, it is very easy to obtain the integrals for any value of m and a. 

The standard normal distribution is obtained by defining the new variable. 

It is very easy to show that the Gaussian given by Eq. 2.65 has mean 

The cumulative standard normal distribution function, Eq. 2j61, is now written as 

---

It is evident that an approximate — 1.5cr shift ean be determined from the data and so the Cpi value is more suitable as a model. Using the graph on Figure 6, whieh shows the relationship Cp, (at 1.5cr shift) and parts-per-million (ppm) failure at the nearest limit, the likely annual failure rate of the produet ean be ealeulated. The figure has been eonstrueted using the Standard Normal Distribution (SND) for various limits. The number of eomponents that would fall out of toleranee at the nearest limit, is potentially 30 000 ppm at = 0.62, that is, 750 eomponents of the 25 000 manufaetured per annum. Of eourse, aetion in the form of a proeess eap-ability study would prevent further out of toleranee eomponents from being produeed and avoid this failure rate in the future and a target Cp = 1.33 would be aimed for. 

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

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... 

A conceptual definition of the follows from consideration of a set of numbers drawn at random from the standard normal distribution, the one with mean zero and standard deviation one. Ordering this set of numbers gives a sequence called order statistics. The are the... 

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). 

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).

Because (a) is the area in the lower tail of the normal distribution, Za is called the ath quantile of the standard normal distribution, (or the (100)(a)th percentile). A useful identity follows directly from the symmetry of the Gaussian distribution in Equation A-2 (4). 

The number 0.128 is the largest acceptable true CV p for which the net error would not exceed +25% at the 95% confidence level. The number 1.96 is the appropiate Z-statistic (from tables of the standard normal distribution) at the same confidence level. 

The 95% critical value from the standard normal distribution for this one-tailed test would be 1.645. Therefore, we would reject the hypothesis of no autocorrelation. 

A common method of simulating random draws from the standard normal distribution is to compute the sum of 12 draws from the uniform [0,1] distribution and subtract 6. Can you justify this procedure ... 

Use Monte Carlo Integration to plot the function g(r) = E[xr x>0] for the standard normal distribution. The expected value from the truncated normal distribution is... 

A similar analysis shows that the variance of X is simply ax- The standard normal distribution is obtained by defining a standardized variable z ... 

To determine the critical region, we must know the distribution of the test statistic. In this case, Z is distributed as the standard normal distribution. With H0 [t<48 and a=0.05, we determine that the critical region will include 5% of the area on the high end of the standard normal curve Fig. 1.6. The Z-value that cuts off 5% of the curve is found to be 1.645, from a table of... 

A few values of the t-distribution are given in an accompanying table. We note that t values are considerably higher than corresponding standard normal values for small sample size but as n increases, the t-distribution asymptotically approaches the standard normal distribution. Even at a sample size as small as 30, the deviation from normality is small, so that it is possible to use the standard normal distribution for sample sizes larger than 30 (n>30) and in most cases, for n<30 t-distribution is used. This is equivalent to assuming that Sx is an exact estimate of ox at large sample sizes (n>30). 

For values k>30, the % -distribution may be approximated from the standard normal distribution ... 

---