Chapter 2.3 - Math3200 Intermediate Probability and Statistics Prof Nan Lin Department of Mathematics Washington University Agenda Discrete random

Chapter 2.3 - Math3200 Intermediate Probability and...

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Math3200 Intermediate Probability and Statistics Prof. Nan Lin Department of Mathematics Washington University

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Agenda Discrete random variables Continuous random variables
Random Variable Random variables are functions that assign a numerical value for every element ? ∈ ? . Denoted using capital letters, e.g., ?, ? Suppose that S is the outcome space for a the result of a baseball game, ? = {?, 𝐿} The random variable ? must translate the domain of S into numerical values E.g. ? ? = ? = 1, ? ? = 𝐿 = 0

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Random variable Suppose S is the outcome space of rolling a die, ? = 1,2,3,4,5,6 , then a naturally defined random variable X will be given by the identity function ? ? = ? . Discrete random variable take on only a countable number of distinct values such as 0,1,2,3,4 ,… Often counts e.g. the number of children in a family, the Friday night attendance at a cinema, the number of defective light bulbs in a box of ten Continuous random variable takes an uncountably infinite number of possible values Often measurements e.g. height, weight, the amount of sugar in an orange, the time required to run a mile
Discrete random variable Probability mass function (p.m.f.) ? 𝑥 = 𝑃 ? = 𝑥 , for 𝑥 ∈ ? Properties ?(𝑥) ≥ 0 for all 𝑥 ∈ ? ?(𝑥) 𝑥∈𝑆 = 1 𝑃 ? ∈ ? = ?(𝑥) 𝑥∈𝐴 ? is also called the ‘support’ of ? Cumulative distribution function (c.d.f.) 𝐹 𝑥 = 𝑃 ? ≤ 𝑥 = ?(?) 𝑘≤𝑥 Nondecreasing 𝐹 −∞ = 0 , 𝐹 +∞ = 1

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Example: discrete uniform distribution ? 𝑥 = 1 𝑚 , for 𝑥 = 1, … , 𝑚 The p.m.f. is constant e.g. roll a fair die 𝑚 = 6
Example: coin tossing Consider an experiment in which a fair coin is tossed twice. Several possible ways to record

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• Spring '08
• SAWYER
• Statistics, Probability, Probability distribution, Probability theory, CDF, continuous random variable

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