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Prove that e²+1 = (1 - nt) ²e + n²

Prove that e²+1 = (1 - nt) ²e + n²

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Probability and Statistics for CS

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Let us assume the following simplified setting: - Assume \( f(\theta) = \theta \). This means that the root is \(\theta^* = 0\). - Assume that \(\eta_i < 1\) for all \(i\). - Assume that \(Z_i\) has a mean of zero and variance of 1. \[ \forall i : E[Z_i] = 0, E[Z_i^2] = 1 \]
Transcribed Image Text:Let us assume the following simplified setting: - Assume \( f(\theta) = \theta \). This means that the root is \(\theta^* = 0\). - Assume that \(\eta_i < 1\) for all \(i\). - Assume that \(Z_i\) has a mean of zero and variance of 1. \[ \forall i : E[Z_i] = 0, E[Z_i^2] = 1 \]
**Prove that** \[ e^2_{t+1} = (1 - \eta_t)^2 e^2_t + \eta^2_t \] This equation represents a mathematical proof challenge. It involves a recursive relation where the squared term \( e^2_{t+1} \) at time \( t+1 \) is expressed as a combination of the squared term \( e^2_t \) at time \( t \), modified by a factor involving \( \eta_t \). To prove this relation, one would typically start by understanding the underlying assumptions and properties of the functions and variables involved, which could include derivatives, initial conditions, or boundary values. This problem could be encountered in a variety of mathematical or scientific fields, including calculus, discrete mathematics, or even signal processing, where understanding recursive relations is fundamental. In proving this, consider possible simplifications or alternative expressions for any terms, and ensure all mathematical operations preserve equality.
Transcribed Image Text:**Prove that** \[ e^2_{t+1} = (1 - \eta_t)^2 e^2_t + \eta^2_t \] This equation represents a mathematical proof challenge. It involves a recursive relation where the squared term \( e^2_{t+1} \) at time \( t+1 \) is expressed as a combination of the squared term \( e^2_t \) at time \( t \), modified by a factor involving \( \eta_t \). To prove this relation, one would typically start by understanding the underlying assumptions and properties of the functions and variables involved, which could include derivatives, initial conditions, or boundary values. This problem could be encountered in a variety of mathematical or scientific fields, including calculus, discrete mathematics, or even signal processing, where understanding recursive relations is fundamental. In proving this, consider possible simplifications or alternative expressions for any terms, and ensure all mathematical operations preserve equality.
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