Chapter 5.3-5.4 - Math3200 Intermediate Probability and Statistics Prof Nan Lin Department of Mathematics Washington University Outline

# Chapter 5.3-5.4 - 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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Outline Student’s ? -distribution Snedecor- Fisher’s 𝐹 -distribution Math3200 Nan Lin, Washington Unviersity 2
Given a random sample ? 1 , … , ? 𝑛 from 𝑁(?, 𝜎 2 ) , we know that ? −? 𝜎/ 𝑛 ∼ 𝑁(0,1) . However, 𝜎 is often unknown in practice. If we replace it by the sample standard deviation ? , what will be the distribution of ? −? 𝑠/ 𝑛 ? The answer is given by the Irish chemist William Sealy Gosset He published his results under the pseudonym “Student” ? − ? ?/ 𝑛 ∼ ? 𝑛−1 Math3200 Nan Lin, Washington Unviersity 3

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Student’s ? -distribution Construction Suppose ?~𝑁(0,1) and ? ∼ 𝜒 ? 2 are independent. Then ? = 𝑍 ?/? ∼ ? ? ? : Degrees of freedom ? −? 𝑠/ 𝑛 = 𝑋 −? 𝜎/ 𝑛 𝑆 2 /𝜎 2 = 𝑋 −? 𝜎/ 𝑛 𝑛−1 𝑆 2 𝜎 2 /(𝑛−1) 𝑁(0,1) 𝜒 𝑛−1 2 /(𝑛−1) Math3200 Nan Lin, Washington Unviersity 4
Tail is heavier than N(0,1) Math3200 Nan Lin, Washington Unviersity

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