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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...
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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.
- Spring '08
- The Land