Homework 4 student.pdf - STAT 350(Spring 2019 Homework 4(26...

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STAT 350 (Spring 2019) Homework 4 (26 points + 1 point BONUS) 1 Practice Problems: 5.1 (p.191), 5.5 (p.191), 5.7 (p.191), Practice Problems: 5.21 (p.200), 5.45 (p.208), 5.47 (p.208) randomized , autograded (1 point) 1. Classify each random variable as discrete or continuous. Please see LON CAPA for the correct answers. (1 point) 2. Show that Var(X) = σ 2 = E(X 2 ) − ( E(X)) 2 . Hint: Use the definition of variance and μ = E(X). Var(?) = ? [(? − ?(?)) 2 ] by definition of variance = ?[(? − ?) 2 ] as ? = ?(?) = ?[? 2 − 2?? + ? 2 ] by multiplication = ?(? 2 ) − 2??(?) + ? 2 by properties of ? and ? being a constant = ?(? 2 ) − 2(?(?)) 2 + (?(?)) 2 as ? = ?(?) = ?(? 2 ) − (?(?)) 2 algebra Practice Problems: 5.21 (p.200), 5.45 (p.208), 5.47 (p.208) Additional Problems: 5.33 (p.201), 5.39abc (p.201), 5.57 (p.209), 5.59 (p.209) (1 point) 3. BONUS: Show using the definition of expected values for discrete random variables (Eq. 5.1) that for a discrete random variable, X, E(g(X )) = ∑ g(x) p(x) where g(x) is linear. That is, show the formula is true assuming that g(x) = ax + b. You are showing that a special case of Eq. 5.2 is true; therefore, you many not use that formula. ?[?(?)] = ?[?? + ?] by the definition of ? = ? ?[?] + ? by properties of expectation = ? ∑ 𝑥 𝑝(𝑥) 𝑥∈? + ? by the definition of expectation for a discrete random variable = ? ∑ 𝑥 𝑝(𝑥) 𝑥∈? + ? ∑ 𝑝(𝑥) 𝑥∈? as all probabilities sum to 1 = ∑ ? 𝑥 𝑝(𝑥) 𝑥∈? + ∑ ? 𝑝(𝑥) 𝑥∈? = ∑(? 𝑥 𝑝(𝑥) + ? 𝑝(𝑥)) 𝑥∈? by properties of summation = ∑(?𝑥 + ?)𝑝(𝑥) 𝑥∈? = ∑ ?(𝑥)𝑝(𝑥) 𝑥∈? . Practice Problems: 5.21 (p.200), 5.45 (p.208), 5.47 (p.208) Additional Problems: 5.33 (p.201), 5.39abc (p.201), 5.57 (p.209), 5.59 (p.209)
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