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b.lect9

# b.lect9 - Outline Parameters of Discrete Distributions...

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Unformatted text preview: Outline Parameters of Discrete Distributions Parameters of Continuous Distributions Lecture 9 Chapter 3: Random Variables and Their Distributions Michael Akritas Michael Akritas Lecture 9 Chapter 3: Random Variables and Their Distributions Outline Parameters of Discrete Distributions Parameters of Continuous Distributions Parameters of Discrete Distributions The Expected Value The Variance Parameters of Continuous Distributions The Expected Value and Variance The Median Other Percentiles (or Quantiles) IQR: Another measure of variability Michael Akritas Lecture 9 Chapter 3: Random Variables and Their Distributions Outline Parameters of Discrete Distributions Parameters of Continuous Distributions The Expected Value The Variance Michael Akritas Lecture 9 Chapter 3: Random Variables and Their Distributions Outline Parameters of Discrete Distributions Parameters of Continuous Distributions The Expected Value The Variance The expected value , E ( X ) or μ X , of a discrete r.v. X having a possibly infinite sample space S X and pmf p ( x ) = P ( X = x ), for x ∈ S X , is defined as μ X = X x in S X xp ( x ) . Michael Akritas Lecture 9 Chapter 3: Random Variables and Their Distributions Outline Parameters of Discrete Distributions Parameters of Continuous Distributions The Expected Value The Variance The expected value , E ( X ) or μ X , of a discrete r.v. X having a possibly infinite sample space S X and pmf p ( x ) = P ( X = x ), for x ∈ S X , is defined as μ X = X x in S X xp ( x ) . To see that this generalizes the definition given in Chapter 1 read Example 3.4.1, Proposition 3.4.1 and Example 3.4.2 in the book. Michael Akritas Lecture 9 Chapter 3: Random Variables and Their Distributions Outline Parameters of Discrete Distributions Parameters of Continuous Distributions The Expected Value The Variance The expected value , E ( X ) or μ X , of a discrete r.v. X having a possibly infinite sample space S X and pmf p ( x ) = P ( X = x ), for x ∈ S X , is defined as μ X = X x in S X xp ( x ) . To see that this generalizes the definition given in Chapter 1 read Example 3.4.1, Proposition 3.4.1 and Example 3.4.2 in the book. Example Roll a die and let X denote the outcome. If X = 1 or 2, you win \$1; if X = 3 you win \$2, and if X ≥ 4 you win \$4. Let Y denote your prize. Find E ( Y ). Michael Akritas Lecture 9 Chapter 3: Random Variables and Their Distributions Outline Parameters of Discrete Distributions Parameters of Continuous Distributions The Expected Value The Variance Example (Continued) Solution: The pmf of Y is: y 1 2 4 p Y ( y ) 0.333 0.167 0.5 Thus, E ( Y ) = 0 . 333 + 2 × . 167 + 4 × . 5 = 2 . 667 Michael Akritas Lecture 9 Chapter 3: Random Variables and Their Distributions Outline Parameters of Discrete Distributions Parameters of Continuous Distributions The Expected Value The Variance Example (Continued) Solution: The pmf of Y is: y 1 2 4 p Y ( y ) 0.333 0.167 0.5 Thus, E ( Y ) = 0 . 333 + 2 × . 167 + 4 × . 5 = 2 . 667 I Sometimes we want to find the expected value of a function...
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b.lect9 - Outline Parameters of Discrete Distributions...

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