Approximate Methods 8. Expand Example A Page 162. EXAMPLE A Theory needed The relation of voltage, current, and resistance is V=IR. Suppose that the voltage is held constant at a value Vo across a medium whose resistance fluctuates randomly as a result, say, of random fluctuations at the molecular level. The current therefore also varies randomly. Suppose that it can be determined experimentally to have mean μ #0 and variance o7. We wish to find the mean and variance of the resistance, R, and since we do not know the distribution of I, we must resort to an approximation. Uy ~ g(ux) oo[g'(x)]² x. To the first order, Y = g(x) = g(x) + (x − µx)g (µx) we can carry out the Taylor series expansion to the second order to get an improvec approximation of μy: Y = g(X) ≈g(µx) + (X − µx)g'(µx) + ½ (X − µx)²g"(µx) Taking the expectation of the right-hand side, we have, since E(X-x) = 0, E(Y) ~g(@x)+jong”(ux)
Approximate Methods 8. Expand Example A Page 162. EXAMPLE A Theory needed The relation of voltage, current, and resistance is V=IR. Suppose that the voltage is held constant at a value Vo across a medium whose resistance fluctuates randomly as a result, say, of random fluctuations at the molecular level. The current therefore also varies randomly. Suppose that it can be determined experimentally to have mean μ #0 and variance o7. We wish to find the mean and variance of the resistance, R, and since we do not know the distribution of I, we must resort to an approximation. Uy ~ g(ux) oo[g'(x)]² x. To the first order, Y = g(x) = g(x) + (x − µx)g (µx) we can carry out the Taylor series expansion to the second order to get an improvec approximation of μy: Y = g(X) ≈g(µx) + (X − µx)g'(µx) + ½ (X − µx)²g"(µx) Taking the expectation of the right-hand side, we have, since E(X-x) = 0, E(Y) ~g(@x)+jong”(ux)
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter3: Straight Lines And Linear Functions
Section3.4: Linear Regression
Problem 20SBE
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Number 8
![Approximate Methods
8. Expand Example A Page 162.
EXAMPLE A
Theory needed
The relation of voltage, current, and resistance is V=IR. Suppose that the voltage
is held constant at a value Vo across a medium whose resistance fluctuates randomly
as a result, say, of random fluctuations at the molecular level. The current therefore
also varies randomly. Suppose that it can be determined experimentally to have mean
μ #0 and variance o7. We wish to find the mean and variance of the resistance, R,
and since we do not know the distribution of I, we must resort to an approximation.
με ~ g(μχ)
of o[g'(x)]²
x. To the first order,
Y = g(X)≈ g(x) + (x − µx)g' (µx)
we can carry out the Taylor series expansion to the second order to get an improvec
approximation of μly:
Y = g(X) ≈g(µx) + (X− µx)g'(µx) + (x − µx)²g" (µx)
Taking the expectation of the right-hand side, we have, since E(X-x) = 0,
E(Y) ~ g(ux)+ jog(x)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80beb3c7-0805-4123-826a-93aa0f72cc1c%2F903433ca-8fa9-4f3e-aa7d-1fbf3ccde25f%2Fua9h1e.jpeg&w=3840&q=75)
Transcribed Image Text:Approximate Methods
8. Expand Example A Page 162.
EXAMPLE A
Theory needed
The relation of voltage, current, and resistance is V=IR. Suppose that the voltage
is held constant at a value Vo across a medium whose resistance fluctuates randomly
as a result, say, of random fluctuations at the molecular level. The current therefore
also varies randomly. Suppose that it can be determined experimentally to have mean
μ #0 and variance o7. We wish to find the mean and variance of the resistance, R,
and since we do not know the distribution of I, we must resort to an approximation.
με ~ g(μχ)
of o[g'(x)]²
x. To the first order,
Y = g(X)≈ g(x) + (x − µx)g' (µx)
we can carry out the Taylor series expansion to the second order to get an improvec
approximation of μly:
Y = g(X) ≈g(µx) + (X− µx)g'(µx) + (x − µx)²g" (µx)
Taking the expectation of the right-hand side, we have, since E(X-x) = 0,
E(Y) ~ g(ux)+ jog(x)
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