Several probability distributions model the outcomes of various trials when the probabilities of certain events are given. For some distributions, the definitions make sense even when a probability is 0. For other distributions, the definitions do not make sense unless all probabilities are strictly positive. This article examines how zero

## Tag: **Simulation**

On this blog, I write about a diverse set of topics that are relevant to statistical programming and data visualization. In a previous article, I presented some of the most popular blog posts from 2021. The most popular articles often deal with elementary or familiar topics that are useful to

You can use the Cholesky decomposition of a covariance matrix to simulate data from a correlated multivariate normal distribution. This method is encapsulated in the RANDNORMAL function in SAS/IML software, but you can also perform the computations manually by calling the ROOT function to get the Cholesky root and then

While discussing how to compute convex hulls in SAS with a colleague, we wondered how the size of the convex hull compares to the size of the sample. For most distributions of points, I claimed that the size of the convex hull is much less than the size of the

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

There are times when it is useful to simulate data. One of the reasons I use simulated data sets is to demonstrate statistical techniques such as multiple or logistic regression. By using SAS random functions and some DATA step logic, you can create variables that follow certain distributions or are

SAS' Bahar Biller, an operations researcher, details how to develop a supply chain digital twin.

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

A previous article shows how to use Monte Carlo simulation to estimate a one-dimensional integral on a finite interval. A larger random sample will (on average) result in an estimate that is closer to the true value of the integral than a smaller sample. This article shows how you can

Numerical integration is important in many areas of applied mathematics and statistics. For one-dimensional integrals on the interval (a, b), SAS software provides two important tools for numerical integration: For common univariate probability distributions, you can use the CDF function to integrate the density, thus obtaining the probability that a

In a previous article, I showed how to generate random points uniformly inside a d-dimensional sphere. In that article, I stated the following fact: If Y is drawn from the uncorrelated multivariate normal distribution, then S = Y / ||Y|| has the uniform distribution on the unit sphere. I was

Imagine an animal that is searching for food in a vast environment where food is scarce. If no prey is nearby, the animal's senses (such as smell and sight) are useless. In that case, a reasonable search strategy is a random walk. The animal can choose a random direction, walk/swim/fly

Do you know that you can create a vector that has a specific correlation with another vector? That is, given a vector, x, and a correlation coefficient, ρ, you can find a vector, y, such that corr(x, y) = ρ. The vectors x and y can have an arbitrary number

When there are two equivalent ways to do something, I advocate choosing the one that is simpler and more efficient. Sometimes, I encounter a SAS program that simulates random numbers in a way that is neither simple nor efficient. This article demonstrates two improvements that you can make to your

The skewness of a distribution indicates whether a distribution is symmetric or not. The Wikipedia article about skewness discusses two common definitions for the sample skewness, including the definition used by SAS. In the middle of the article, you will discover the following sentence: In general, the [estimators] are both

The triangulation theorem for polygons says that every simple polygon can be triangulated. In fact, if the polygon has V vertices, you can decompose it into V-2 non-overlapping triangles. In this article, a "polygon" always means a simple polygon. Also, a "random point" means one that is drawn at random

How can you efficiently generate N random uniform points in a triangular region of the plane? There is a very cool algorithm (which I call the reflection method) that makes the process easy. I no longer remember where I saw this algorithm, but it is different from the "weighted average"

The Poisson-binomial distribution is a generalization of the binomial distribution. For the binomial distribution, you carry out N independent and identical Bernoulli trials. Each trial has a probability, p, of success. The total number of successes, which can be between 0 and N, is a binomial random variable. The distribution

In the paper "Tips and Techniques for Using the Random-Number Generators in SAS" (Sarle and Wicklin, 2018), I discussed an example that uses the new STREAMREWIND subroutine in Base SAS 9.4M5. As its name implies, the STREAMREWIND subroutine rewinds a random number stream, essentially resetting the stream to the beginning.

A previous article about standardizing data in groups shows how to simulate data from two groups. One sample (with n1=20 observations) is simulated from an N(15, 5) distribution whereas a second (with n2=30 observations) is simulated from an N(16, 5) distribution. The sample means of the two groups are close

Books about statistics and machine learning often discuss the tradeoff between bias and variance for an estimator. These discussions are often motivated by a sophisticated predictive model such as a regression or a decision tree. But the basic idea can be seen in much simpler situations. This article presents a

In a previous article, I discussed the binormal model for a binary classification problem. This model assumes a set of scores that are normally distributed for each population, and the mean of the scores for the Negative population is less than the mean of scores for the Positive population. I