# 10 - BTRY/STSCI 4080 Homework 10 Due Date before noon on...

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Unformatted text preview: BTRY/STSCI 4080 Homework # 10 Due Date: before noon on 12/6/10 Problem numbers beginning with P (e.g., P1) denote problems from the “PROB- LEMS” section; those beginning with T (e.g., T1) denote problems taken from the “THEORETICAL EXERCISES” section. 1. Let a,b ∈ R , with a < b . Suppose g ( · ) is a function that satisfies: • g ( u ) > 0 and is continuous for u ∈ [ a,b ] • max u ∈ [ a,b ] g ( u ) = M < ∞ (i.e., g ( u ) is bounded between [0 ,M ] for u ∈ [ a,b ]). Let X 1 ,...,X n be independent Uniform( a,b ) random variables. Let Y 1 ,...,Y n be indepen- dent Uniform(0 ,M ) random variables. Suppose the X ’s and Y ’s are all mutually independent. Finally, define Z i = I { g ( X i ) > Y i } , and W n = M ( b- a ) n ∑ n i =1 Z i . (a) Explain why Z 1 ...Z n are independent Bernoulli( p ) random variables, where p = P ( g ( X 1 ) > Y 1 ) . What is the probability distribution of ∑ n i =1 Z i ? (b) Show that P ( g ( X 1 ) > Y 1 ) = 1 M ( b- a ) b a g ( x ) dx . (c) Show that E ( W n ) = I g and V ar ( W n ) = n- 1 I g ( M ( b- a )-I g ), where I g = b a g ( u ) du. (d) Let g ( · ) = φ ( · ), where φ ( · ) is the standard normal pdf. Consider the integral I g = 1 . 28- 1 . 645 φ ( u ) du = 0 . 8497425 . Show that one can take M = φ (0) = 1 √ 2 π and compute V ar ( W n ) . How large must n be to ensure that SD ( W n ) < . 001?...
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10 - BTRY/STSCI 4080 Homework 10 Due Date before noon on...

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