Standard normal table

Abstract: This article introduces a class of distortion operators, ga(t) = D[44-(u) + a], where D is the standard normal cumulative distribution For any loss (or asset) variable X with a probability distribution Sx(x) = 1Fx(x), ga [Sx(x)] defines a distorted probability distribution whose mean value yields a risk-adjusted premium (or an asset price) The distortion operator ga can be applied to both assets and liabilities, with opposite signs in the parameter a Based on CAPM, the author establishes that the parameter ca should correspond to the systematic risk of X For a normal (L,aU2) distribution, the distorted distribution is also normal with '= u + aa and a5' = a For a lognormal distribution, the distorted dis

Abstract: When we measure a quantity in a large number of individuals we call the pattern of values obtained a distribution. For example, figure 1 shows the distribution of serum albumin concentration in a sample of adults displayed as a histogram. This is an empirical distribution. There are also theoretical distributions, of which the best known is the normal distribution (sometimes called the Gaussian distribution), which is shown in figure 2. Although widely referred to in statistics, the normal distribution remains a mysterious concept to many. Here we try to explain what it is and why it is important. FIG 3 (left)—Serum albumin values in 248 adults FIG 2 (right)—Normal distribution with the same mean and standard deviation as the serum albumin values In this context the name “normal” causes much confusion. In statistics it is just a name; …

Abstract: The usual statistical technique used to compare the means of two groups is a confidence interval or significance test based on the t distribution. For this we must assume that the data are samples from normal distributions with the same variance. Table 1 shows the biceps skinfold measurements for 20 patients with Crohn's disease and nine patients with coeliac disease. View this table: Table 1 Biceps skinfold thickness (mm) in two groups of patients The data have been put into order of magnitude, and it is fairly obvious that the distribution is skewed and far from normal. When, as here, the assumption of normality is wrong we can often transform the …

Abstract: When outcomes are binary, the c-statistic (equivalent to the area under the Receiver Operating Characteristic curve) is a standard measure of the predictive accuracy of a logistic regression model. An analytical expression was derived under the assumption that a continuous explanatory variable follows a normal distribution in those with and without the condition. We then conducted an extensive set of Monte Carlo simulations to examine whether the expressions derived under the assumption of binormality allowed for accurate prediction of the empirical c-statistic when the explanatory variable followed a normal distribution in the combined sample of those with and without the condition. We also examine the accuracy of the predicted c-statistic when the explanatory variable followed a gamma, log-normal or uniform distribution in combined sample of those with and without the condition. Under the assumption of binormality with equality of variances, the c-statistic follows a standard normal cumulative distribution function with dependence on the product of the standard deviation of the normal components (reflecting more heterogeneity) and the log-odds ratio (reflecting larger effects). Under the assumption of binormality with unequal variances, the c-statistic follows a standard normal cumulative distribution function with dependence on the standardized difference of the explanatory variable in those with and without the condition. In our Monte Carlo simulations, we found that these expressions allowed for reasonably accurate prediction of the empirical c-statistic when the distribution of the explanatory variable was normal, gamma, log-normal, and uniform in the entire sample of those with and without the condition. The discriminative ability of a continuous explanatory variable cannot be judged by its odds ratio alone, but always needs to be considered in relation to the heterogeneity of the population.