Recall that the binomial distribution is the distribution of the number of successes in a set of independent Bernoulli trials, each having the same probability of success. Most introductory statistics textbooks discuss the approximation of the binomial distribution by the normal distribution. The graph to the right shows that the
The field of probability and statistics is full of asymptotic results. The Law of Large Numbers and the Central Limit Theorem are two famous examples. An asymptotic result can be both a blessing and a curse. For example, consider a result that says that the distribution of some statistic converges
A statistical programmer asked how to simulate event-trials data for groups. The subjects in each group have a different probability of experiencing the event. This article describes one way to simulate this scenario. The simulation is similar to simulating from a mixture distribution. This article also shows three different ways
In general, it is hard to simulate multivariate data that has a specified correlation structure. Copulas make that task easier for continuous distributions. A previous article presented the geometry behind a copula and explained copulas in an intuitive way. Although I strongly believe that statistical practitioners should be familiar with
Do you know what a copula is? It is a popular way to simulate multivariate correlated data. The literature for copulas is mathematically formidable, but this article provides an intuitive introduction to copulas by describing the geometry of the transformations that are involved in the simulation process. Although there are
This article uses simulation to demonstrate the fact that any continuous distribution can be transformed into the uniform distribution on (0,1). The function that performs this transformation is a familiar one: it is the cumulative distribution function (CDF). A continuous CDF is defined as an integral, so the transformation is
A previous article showed how to simulate multivariate correlated data by using the Iman-Conover transformation (Iman and Conover, 1982). The transformation preserves the marginal distributions of the original data but permutes the values (columnwise) to induce a new correlation among the variables. When I first read about the Iman-Conover transformation,
Simulating univariate data is relatively easy. Simulating multivariate data is much harder. The main difficulty is to generate variables that have given univariate distributions but also are correlated with each other according to a specified correlation matrix. However, Iman and Conover (1982, "A distribution-free approach to inducing rank correlation among
Quick! Which fraction is bigger, 40/83 or 27/56? It's not always easy to mentally compare two fractions to determine which is larger. For this example, you can easily see that both fractions are a little less than 1/2, but to compare the numbers you need to compare the products 40*56
As mentioned in my article about Monte Carlo estimate of (one-dimensional) integrals, one of the advantages of Monte Carlo integration is that you can perform multivariate integrals on complicated regions. This article demonstrates how to use SAS to obtain a Monte Carlo estimate of a double integral over rectangular and
