Normal distribution with known variance

Single Draw from a Normal Distribution with Known Variance... 

Exact analytical solutions can be obtained for summations of normal distributions. The sum of normal distributions is identically a normal distribution. The mean of the sum is the sum of the means of each input distribution. The variance of the sum is the sum of the variance of the inputs. Any statistic of interest for the output can be estimated by knowing its distribution type and its parameters. For example, for a model output that is a normal distribution with known parameter values, one can estimate the 95th percentile of that output. 

Random sample from a normal distribution with krumn variance. Suppose j/i,..., are a random sample from a normal p,a ) distribution where the variance is a known constant. For a random sample, the joint likelihood is the product of the individual likelihocxls, so it is given by... 

Example 16 (continued) We continue with our example where we have a single random drawn from each of j = 1,..., J normally distributed populations where each population has its own mean and its own known variance. The observation yj comes from a normal pj, Oj) distribution with known variance aj. The population distributions are related. We model this relationship between the populations by considering the population means p, ..., pj to be random draws from a normal T, ip)... 

The main difference between the Z-test and the /-test is that the Z-statistic is based on a known standard deviation, a, while the /-statistic uses the sample standard deviation, s, as an estimate of a. With the assumption of normally distributed data, the variance sample variance, v2 as n gets large. It can be shown that the /-test is equivalent to the Z-test for infinite degrees-of-freedom. In practice, a large sample is usually considered n > 30. 

Even with powerful computer programs at hand, the solution of estimation problems is usually far from simple. A convenient way to eliminate computational errors and to study the effects of statistical assumptions is to solve first a problem with known true parameter values, involving data generated at some nominal parameter vector. Initially it is advisable to investigate with error-free data, then to add errors of the assumed structure. The simulation usually requires normally distributed random variables. Random numbers R that approximately are from a normal distribution with zero mean and unit variance can be obtained by... 

Lets assume that errors e, and 8, are independent values which have normal distributions with mean values equal to zero, and variances o and o, correspondingly. In this case, if ox is known as the standard uncertainty of the measurement standard (CRM), the slope (P,) and the intercept (P0) of the calibration curve can be estimated by the following equations [6] ... 

The observation was regarded as normally distributed, with a known variance cr bility density... 

Global Two-Stage Method. An extensive description of the method is provided by Steimer et al. The global two-stage (GTS) approach has been shown, through simulation, to provide unbiased estimates of the population mean parameters and their variance-covariance, whereas the estimates of the variances were upwardly biased if the STS approach was used. These simulations were done under the ideal situation that the residual error was normally distributed with a known variance. However, it is a well-known fact that the asymptotic covariance matrix used in the calculations is approximate and under less ideal conditions, the approximation can be poor. ... 

Normal distribution theory can be used to test whether a particular sample value is consistent with other values or with our past experience. If the mean p and the variance are known, then we can determine how deviant an observed value x,-, appears to be by calculating the statistic Z = (x,- - p)/cr and comparing this with the table ofstandard normal deviates. Suppose that one of the values for our QC specimen was 170ng/ml. Past experience has led us to believe that the results for this QC specimen are normally distributed with p = 207.6 ng/ml and cr = 14.1. Is the... 

When d,j is independent of 5,y, it is known [8] that A( ) obeys a normal distribution with mean [(A — l)3 — (N — l)]/6 and variance (N — l)N2 (N — 2)2/36. Therefore, if the probability of A( j under this null hypothesis is small enough (e.g., less than 0.5%), we can reject this hypothesis and confirm that dij is related to 5,y. This means that object i is placed at a suitable location in the embedded space with a certain confidence level. 

Assume that the random variable X has a normal distribution with an unknown population mean, p, and with a known population variance, For a sample size of n, the sampling distribution of the sample mean has a normal distribution with population mean, p, and variance, The... 

Figure A.6 Line plot of likelihood function (solid line) and log-likelihood function (dashed line) assuming a normal distribution with a known variance equal to 1 and given the data Y = —2.0, — 1.6, — 1.4, 0.25, 0, 0.33, 0.5, 1, 2, and 3. ...

Consider the data shown in Figure 9.13 with two unknown parameters and only 10 data points. The measurement errors are drawn from a normal distributed with zero mean and variance rr = 10 Compute the best estimates of activation energy and mean rate constant and the 95% confidence Intervals for the cases of known. and unknown measurement variance. 

This example is actually unrealistic in two ways. First, we don t run trials with two experimental treatments unless they are different doses of the same treatment, which is not the case here and second, for reasons discussed in Chapter 3, we are generally interested in controlled comparisons and not population averages. However, to avoid some complications we shall accept the example as described here with the further simplification that we will assume that the variance of individual responses is known and the same for each treatment.) We suppose that the sample mean for a given treatment i is and is Normally distributed with variance erf /n. 

Thus, there are two parameters in the model. Fiuthermore, let m have a normal distribution with mean zero and variance and suppose that the true value of is known. This effectively makes om model have only one parameter, m. We can plug these models into the Bayes mle machinery, as embodied in Eq. (8.2) some algebraic calculations would reveal that the posterior distribution of m given the data has a closed-form solution. It is normal with mean nP-x/ nP- - - i ) and variance nf- + In the mean, the bar over the x denotes the average of the observations. 

The first result here is known as the elementary renewal theorem. Simply put, it says that if the average time between arrivals is a half-hour, then on average there wiU be two arrivals per hour (A = 2) in the long run. To enhance the results (2.18), we may invoke an extension of the central limit theorem of Subsection 1.3, stating that N f) is asymptotically normally distributed with mean At and variance XcH for large times t. This fact is useM in understanding queues in heavy traffic see Subsection 5.2. 

Measurements y t,0) obtained by n sensors from the real system carry some uncertainty. Therefore, the residuals e t, 0) = y t, 0) — y(t, 0) between measured and computed vectors, the cost function and the estimated parameters 0 are uncertain. Each residual ej 0) = e tk-q- +j, 0) has an error Sj. It is assumed that all these errors ej, j = 1,..., q + 1, are independent and that each of its n components is normally distributed with mean zero and a known variance i = 1,..., n. If the known variances are quite different, a weighted least squares problem may be considered. 

If the measurement is carried out in a way preventing die occurrence of systematic errors, then it is often reasonable to assume that the errors have normal distribution with zero mean and known variance and can be estimated by analysis of the experimental procedure. According to the maximum-likelihood method (77BEC1) the most probable values of the parameters are i... 

When the population distributirai is known, for example X is normally distributed with mean p and variance o, we may calculate the probability of X being equal to or larger than x (X > x). We use that principle for e.g. process control by control charts. The production process has known targets for p and and when an uncommon value of variable X > x is observed, the production process should be adjusted. 

NORMALLY DISTRIBUTED OBSERVATIONS WITH KNOWN VARIANCE 75... 

The measurement error, V-, is often assumed to be a normalized, normally distributed random variable with a zero mean and a known variance. The normal probability density function for the measurement error at condition i is... 

Here the data y, the )th observation for the ith subject, are assumed to be known and independently normally distributed around the model prediction /(, Xt,) with variance d. 9i represents a vector of individual parameter values for the /th individual and Xij is the sampling time. 

When the population variance is not known, the confidence interval for a population mean cannot be determined using a standard normal distribution, but it can be determined with the t distribution. As discussed in Section 3.2.5, one defines the test statistic t in terms of x, p, n, and the sample variance (see Equation (3.13)). By replacing the generic test statistic y in Equation (3.17) with t, the interval for p is defined [2] by... 

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