=)²dx<∞. Let U₁, U2,... be IID Unif(0, 1) random variables. F 1 In = Σƒ(U). i=1 Argue that µ = E[ƒ(U;)] = I and o² = Var[ƒ(U₁)] = fo f(x)²dx Assume that o ≤ 1. How large should n be taken so that wit CTO
=)²dx<∞. Let U₁, U2,... be IID Unif(0, 1) random variables. F 1 In = Σƒ(U). i=1 Argue that µ = E[ƒ(U;)] = I and o² = Var[ƒ(U₁)] = fo f(x)²dx Assume that o ≤ 1. How large should n be taken so that wit CTO
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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Question
8
![8. Let f be a function defined on [0, 1]. Put I = f f(x)dx and assume that
f f(x)²dx <∞. Let U₁, U₂, be IID Unif(0, 1) random variables. For each
n, let
n
1
In
f(U).
(a) Argue that μ = E[ƒ(U;)] = I and o²
] = I and o² = Var[ƒ (U;)] = fő ƒ (x)²dx – 1².
(b) Assume that o≤ 1. How large should n be taken so that with 95%
confidence, the estimate In, is within 0.01 of the true value of I?
(c) Compare this with what can be deduced using only Chebyshev's In-
equality (as was done in Lecture 19).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9201dc12-d7ad-491a-b878-e567deafd5de%2F8aeca585-d3e8-4b99-91fd-9971a68a9714%2Fhpqssoq_processed.png&w=3840&q=75)
Transcribed Image Text:8. Let f be a function defined on [0, 1]. Put I = f f(x)dx and assume that
f f(x)²dx <∞. Let U₁, U₂, be IID Unif(0, 1) random variables. For each
n, let
n
1
In
f(U).
(a) Argue that μ = E[ƒ(U;)] = I and o²
] = I and o² = Var[ƒ (U;)] = fő ƒ (x)²dx – 1².
(b) Assume that o≤ 1. How large should n be taken so that with 95%
confidence, the estimate In, is within 0.01 of the true value of I?
(c) Compare this with what can be deduced using only Chebyshev's In-
equality (as was done in Lecture 19).
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