b.lect10 - Outline Models for Continuous RVs – The Normal...

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Unformatted text preview: Outline Models for Continuous RVs – The Normal Distribution Lecture 10 Chapter 3: Random Variables and Their Distributions M. George Akritas M. George Akritas Lecture 10 Chapter 3: Random Variables and Their Distribution Outline Models for Continuous RVs – The Normal Distribution Models for Continuous RVs – The Normal Distribution Definition: The pdf and cdf Finding Probabilities via the Standard Normal Table Finding Percentiles via the Standard Normal Table M. George Akritas Lecture 10 Chapter 3: Random Variables and Their Distribution Outline Models for Continuous RVs – The Normal Distribution Definition: The pdf and cdf Finding Probabilities via the Standard Normal Table Finding Percentiles via the Standard Normal Table I The Normal distribution if the most important distribution in probability and statistics. I X ∼ N ( μ,σ 2 ) if its pdf is f ( x ; μ,σ 2 ) = 1 √ 2 πσ 2 e- ( x- μ ) 2 2 σ 2 ,-∞ < x < ∞ . I The cdf, F ( x ; μ,σ ), does not have a closed form expression. I R command for f ( x ; μ,σ 2 ): dnorm(x, μ , σ ). I For example, dnorm(0,0,1) gives 0.3989423, which is the value of f (0; μ = 0 , σ 2 = 1). I R command for F ( x ; μ,σ 2 ): pnorm(x, μ , σ ). I For example, pnorm(0,0,1) gives 0.5, which is the value of F (0; μ = 0 , σ 2 = 1). M. George Akritas Lecture 10 Chapter 3: Random Variables and Their Distribution Outline Models for Continuous RVs – The Normal Distribution Definition: The pdf and cdf Finding Probabilities via the Standard Normal Table Finding Percentiles via the Standard Normal Table The Standard Normal Distribution When μ = 0 and σ = 1, X is said to have the standard normal distribution and is denoted, universally, by Z . The pdf of Z is φ ( z ) = 1 √ 2 π e- z 2 / 2 ,-∞ < z < ∞ . The cdf of Z is denoted by Φ. Thus Φ( z ) = P ( Z ≤ z ) = Z z-∞ φ ( x ) dx . Φ( z ) has no closed form expression, but is tabulated in Table A.3 M. George Akritas Lecture 10 Chapter 3: Random Variables and Their Distribution Outline Models for Continuous RVs – The Normal Distribution Definition: The pdf and cdf Finding Probabilities via the Standard Normal Table Finding Percentiles via the Standard Normal Table Plot of φ ( z ) f(x)-3-2-1 1 2 3 0.0 0.1 0.2 0.3 0.4 mu=0, sigm^2=1 mu M. George Akritas Lecture 10 Chapter 3: Random Variables and Their Distribution Outline Models for Continuous RVs – The Normal Distribution...
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This note was uploaded on 01/11/2012 for the course STAT 401 taught by Professor Akritas during the Fall '00 term at Penn State.

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b.lect10 - Outline Models for Continuous RVs – The Normal...

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