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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)

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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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)
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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