# Likelihood ratio test expectation of a random

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: trengthened and many calculations made by appealing to the law of total probability and the deﬁnition of conditional probability. In particular, we sometimes look back, conditioning on what happened at the end of some scenario, and ask what is the conditional probability that the observation happened in a particular way–using Bayes rule. Binomial coeﬃcients form a link between counting and the binomial probability distribution. A small number of key discrete-type and continuous-type distributions arise again and again in applications. Knowing the form of the CDFs, pdfs, or pmfs, and formulas for the means and variances, and why each distribution arises frequently in nature and applications, can thus lead to eﬃcient modeling and problem solving. There are relationships among the key distributions. For example, the binomial distribution generalizes the Bernoulli, and the Poisson distribution is the large n, small p limit of the Bernoulli distribution with np = λ. The exponential distribution is the continuous time version of the geometric distribution; both are memoryless. The exponential distribution is the limit of scaled geometric distributions, and the Gaussian (or normal) distribution, by the central limit theorem, is the limit of standardized sums of large numbers of independent, identically distributed random variables. The following important concepts apply to both discrete-type random variables and continuoustype random variables: • Independence of random variables • Marginals and conditionals 173 174 CHAPTER 5. WRAP-UP...
View Full Document

## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

Ask a homework question - tutors are online