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This problem explores some properties of Z~ N(0, 1). Recall that the pdf of Z is denoted by and the cdf by Þ. (a) Find the value of [*(1-4 (√t)) dt. (b) Consider the function f with domain (-∞, +∞) and range (0, +∞) such that f(u) = E(Z - u). Find the global minimizer and the minimum value of f. (c) This part consists of four subparts. (i) Obtain an expression for o(¹1), the first derivative of o, in terms of . (ii) Explain, in detail, why (¹) is a continuous function on ( (iii) Fix a > 0 arbitrarily. Show that Justify your answer. (iv) Show, in detail, that [*~*xp(x)dx = o(a). lim exp a4+x (²7²) P(|Z| > a) = 0. ∞, +∞).

This problem explores some properties of Z~ N(0, 1). Recall that the pdf of Z is denoted by and the cdf by Þ. (a) Find the value of [*(1-4 (√t)) dt. (b) Consider the function f with domain (-∞, +∞) and range (0, +∞) such that f(u) = E(Z - u). Find the global minimizer and the minimum value of f. (c) This part consists of four subparts. (i) Obtain an expression for o(¹1), the first derivative of o, in terms of . (ii) Explain, in detail, why (¹) is a continuous function on ( (iii) Fix a > 0 arbitrarily. Show that Justify your answer. (iv) Show, in detail, that [*~*xp(x)dx = o(a). lim exp a4+x (²7²) P(|Z| > a) = 0. ∞, +∞).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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This problem explores some properties of Z~ N(0, 1). Recall that the pdf of Z is denoted by and the cdf by Þ. (a) Find the value of [*(1-(√₁)) dt. (b) Consider the function f with domain (-∞, +∞) and range (0, +∞) such that f(u) = E(Z - u). Find the global minimizer and the minimum value of f. (c) This part consists of four subparts. (i) Obtain an expression for o(¹1), the first derivative of o, in terms of . (ii) Explain, in detail, why (¹) is a continuous function on (-∞, +∞). (iii) Fix a > 0 arbitrarily. Show that T. [** xo(x)dx = o(a). Justify your answer. (iv) Show, in detail, that lim exp p(27²)P(|Z| > a) = a11x = 0.
Transcribed Image Text:This problem explores some properties of Z~ N(0, 1). Recall that the pdf of Z is denoted by and the cdf by Þ. (a) Find the value of [*(1-(√₁)) dt. (b) Consider the function f with domain (-∞, +∞) and range (0, +∞) such that f(u) = E(Z - u). Find the global minimizer and the minimum value of f. (c) This part consists of four subparts. (i) Obtain an expression for o(¹1), the first derivative of o, in terms of . (ii) Explain, in detail, why (¹) is a continuous function on (-∞, +∞). (iii) Fix a > 0 arbitrarily. Show that T. [** xo(x)dx = o(a). Justify your answer. (iv) Show, in detail, that lim exp p(27²)P(|Z| > a) = a11x = 0.
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