Nonnal distribution

Source of Nonnal distribution parameters Mean, /i Standard deviation, a... 

The reliability function corresponding to nonnally distributed failure time is given by... 

Tlie nonnal distribution is used to obtain probabilities concerning tlie mean X of a sample of n observations on a random variable X. If X is nonnally distributed witli mean p and standard deviation a, tlien X, tlie sample mean, is nonnally distributed witli mean p. and standard deviation. For example, suppose X is nonnally distributed witli mean 100 and standard deviation 2. 

A nonnegative random variable X has a log-nonnal distribution whenever In X, the natural logariilun of X, has a normal distribution. The pdf of a random variable x liaving a log-normal distribution is specified by... 

Figure 20.5.3 plots tlie pdf of the log-noniial distribution for a = 0 and (3=1. Probabilities concerning random variables liaving a log-normal distribution can be calculated using tables of the normal distribution. If X lias a log-normal distribution witli parameters a and p, then In X lias a normal distribution with p = a and o = p. Probabilities concerning X can tlierefore be converted into equivalent probabilities concerning In X. Suppose, for example, tliat X lias a log-nonnal distribution with a = 2 and p = 0.1. Then... 

Mosleh, Kazarians, and Gekler obtained a Bayesian estimate of the failure rate, Z, of a coolant recycle pump in llie hazard/risk study of a chemical plant. The estimate was based on evidence of no failures in 10 years of operation. Nuclear industry experience with pumps of similar types was used to establish tire prior distribution of Z. Tliis experience indicated tliat tire 5 and 95 percentiles of lire failure rate distribution developed for tliis category were 2.0 x 10" per hour (about one failure per 57 years of operation) and 98.3 x 10 per hour (about one failure per year). Extensive experience in other industries suggested the use of a log-nonnal distribution witli tlie 5 and 95 percentile values as llie prior distribution of Z, tlie failure rate of the coolant recycle pump. 

Recall lliat if Z has a log-nonnal distribution, tlien In Z lias a nonnal distribution with mean p and standard deviation g equal to the parameters in tlie pdf of Z. This fact, in conjunction witli tlie specified 5 and 95 percentiles of llie probability distribution of Z, can be used to obtain the values of p and a, and tliereby the parameters of tlie log-nonnal pdf of Z. If Z denotes the failure rate per year, tlie fact tliat tlie 5" percentile of the distribution of Z is 1/57 implies... 

Derive the log-likelihood function for the model in (13-18) assuming that, it and u, are normally distributed. [Hints Write the log-likelihood function as InL = Z "=1 InLj where InT, is the log-likelihood function for the T observations in group i. These T observations are joint nonnally distributed with covariance matrix given in (14-20).] The log-likelihood is the sum of the logs of the joint normal densities of the n sets of T observations,... 

Suppose that xx and x2 are distributed as independent standard normal. What is the joint distribution of Vi = 2 + 3x, + 2x2 and y2 = 4 + 5xi Suppose you were able to obtain two samples of observations from independent standard normal distributions. How would you obtain a sample from the bivariate nonnal distribution with means 1 and 2 variances 4 and 9 and covariance 3 ... 

Continuation orthogonal polynomials. The Hermite polynomials are orthogonal if x has a standard nonnal distribution. That is, E HHj[ = 0 if i j. Prove this for the Hh H2, and H3 which you obtained above. 

The fourth moment of the standard nonnal distribution is 3 times the variance. Finally,... 

Suppose that the following sample is drawn from a nonnal distribution with mean u and standard deviation ct y = 3.1, -.1,. 3, 1.4, 2.9,. 3, 2.2, 1.5, 4.2,. 4. Test the hypothesis that the mean of the distribution which produced these data is the same as that which produced the data in Exercise 1. Test the hypothesis assuming that the variances are the same. Test the hypothesis that the variances are the same using an F test and using a likelihood ratio test. (Do not assume that the means are the same.)... 

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

If X is not nonnally distributed, tlien X, tlie mean of a sample of n observations on X, is approximately normally distributed witli mean and standard deviation, provided tlie sample size n is large (>30). Tltis result is based on an... 

If X is tlie mean of a random sample of 48 observations on X, X is approximately nonnally distributed with mean 1 and standard deviation... 

Based upon the properties of the nonnal distribution the following values for Equation (65) can be calculated ... 

If we have a set of nonnally distributed measured data, [pp y, ., yn], their... 

FIG. 9.11. Nonnalized distribution ratio of Np(IV) complex between chloroform and an aqueous HSO/ solution. (From Sullivan and Hindman.)... 

Contiruring to use the data in Exercise 1, consider, once again, only the nonzero observations. Suppose that the sampling mechanism is as follows y and another nonnally distributed random variable, z, have population correlation 0.7. The two variables, y and z are sampled jointly. When z is greater than zero, y is reported. When z is less than zero, both z and y are discarded. Exaetly 35 draws were required in order to obtain the preceding sample. Estimate p and a. [Hint Use Theorem 20.4.]... 

Fig. A.9 Nonnal distribution probability density for a continuously variable quantity.

Equation 3.3 is the basis of the theoretical calculations of global polymer properties which depend on chain conformation. For example, the probability density (nonnalized distribution functicxi) G(R) that the chain has a specified end-to-end distance R is expressed in terms of P( r ) as follows ... 

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