Standardized normal variable

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... 

From Table 20.6.2, z, is found to be -1.28. The corresponding simulated value of Ta is obtained by noting tliat (Ta - 100)/20 is a standard normal variable. Therefore, tlie simulated value of Ta corresponding to z, = -1.28 is... 

Since (In Z - p)/a is a standard normal variable, we can refer to a table tliat gives areas under tlie standard nonnal curve (see Table 21.5.2) to find tliat... 

When discussing normally distributed variables, a convenient way of expressing the data is the distance from the mean in standard deviations. This is known as the standardized normal variable or z value. A result that has z = -1.2 is 1.2 times a less than the mean. 

Suppose x and x2 have the bivariate normal distribution described in Section 3.8. Consider an extension of Example 3.4, where the bivariate normal distribution is obtained by transforming two independent standard normal variables. Obtain the distribution of z exp(yi)exp(y2) where y and y2 have a bivariate normal distribution and are correlated. Solve this problem in two ways. First, use the transformation approach described in Section 3.6.4. Second, note that z exp(yl+y2) = exp(vv), so you can first find the distribution of w, then use the results of Section 3.5 (and, in fact, Section 3.4.4 as well). 

Chi-square x 1 exp(—-) /( )- k 2 0 C-) k 2k Distribution of a sum of squares of independent standard normal variables, k is referred to as degrees of freedom Statistical tests on assumed normal distribution. 

Since we are using a normal approximation, the test statistic is simply the standard normal variable ... 

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 ... 

Because x is the square of standard normal variables, it has no negative values ... 

Equation (1.90) gives values that are approximately distributed as the standard normal variable. The x distribution will be used later for tests on the variance, because the following statistic has the x distribution with k degrees of freedom as shown by Brownlee [10],... 

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,... 

First determine the values of the standard normal variable, Z, for component A using the 10 random numbers given in the problem statement and a standard normal table. Then calculate the lifetime of thermometer component A, using the equation for T. 

Determine the values of the standard normal variable and the lifetime of the thermometer component for component C. 

Since C is normally distributed with mean m = 100 and standard deviation <7 = 2, then (C — 100)/2 is a standard normal variable and... 

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. 

Exercise 2.13. What is the variance of the distribution of the difference xi — %2, where xj andx are two standardized normal variables that are perfectly positively... 

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