Question

Let R denote the resistance of a resistor that is selected at random from a population of resistors that are labeled 100Ω. The true population mean resistance is μR = 100 Ω, and the population standard deviation is σR = 2 Ω. The resistance is measured twice with an ohmmeter. Let M1 and M2 denote the measured values. Then M1 = R + E1 and M2 = R + E2, where E1 and E2 are the errors in the measurements. Suppose that E1 and E2 are random with μE1 = μE2 = 0 and σE1 = σE2 = 10Ω. Furthermore, suppose that E1, E2, and R are independent. a. Find σM1 and σM2. b. Show that μM1M2 = μR^2. c. Show that μM1μM2 = μ^2R. d. Use the results of (b) and (c) to show that Cov(M1, M2) = σ^2R. e. Find ρM1,M2.

          Let R denote the resistance of a resistor that is selected at random from a population of resistors that are labeled 100Ω. The true population mean resistance is μR = 100 Ω, and the population standard deviation is σR = 2 Ω. The resistance is measured twice with an ohmmeter. Let M1 and M2 denote the measured values.
Then M1 = R + E1 and M2 = R + E2, where E1 and E2 are the errors in the measurements. Suppose that E1 and E2 are random with μE1 = μE2 = 0 and σE1 = σE2 = 10Ω. Furthermore, suppose that E1, E2, and R are independent.
a. Find σM1 and σM2.
b. Show that μM1M2 = μR^2.
c. Show that μM1μM2 = μ^2R.
d. Use the results of (b) and (c) to show that Cov(M1, M2) = σ^2R.
e. Find ρM1,M2.

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Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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10/02/2022

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Let R denote the resistance of a resistor that is selected at random from a population of resistors that are labeled 100Ω. The true population mean resistance is μR = 100 Ω, and the population standard deviation is σR = 2 Ω. The resistance is measured twice with an ohmmeter. Let M1 and M2 denote the measured values. Then M1 = R + E1 and M2 = R + E2, where E1 and E2 are the errors in the measurements. Suppose that E1 and E2 are random with μE1 = μE2 = 0 and σE1 = σE2 = 10Ω. Furthermore, suppose that E1, E2, and R are independent. a. Find σM1 and σM2. b. Show that μM1M2 = μR^2. c. Show that μM1μM2 = μ^2R. d. Use the results of (b) and (c) to show that Cov(M1, M2) = σ^2R. e. Find ρM1,M2.

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Transcript

00:01 Hi, i'm david and i'm here.

00:02 Now let me bring up your question here.

00:06 In the question here, let me summarize the question where we have the r, it will have the mean r, it will equal to 100 om, and the standard division of the r equal to the 2 om.

00:21 And we are given the e1 will be the error measurement with the mew e1.

00:30 E1 it will equal to the zero and the standard division of the e1 equal to the 10 similarly we have the e2 it will be the same thing where we will have the mu of the e2 it will equal to the 0 as well and the standard division e2 equal to the 10 and we have the e1, e2 and r are independent.

01:00 The r, e1, e2 are independent.

01:08 Now we define the random variable m1 equal to the r plus the e1 and then the m2 equal to the r plus the e2.

01:21 And now the first question a is going to find the sigma of the m1.

01:28 So the sigma of the m1 just equal to the square root of the variance of the m1, which is equal to the r plus the e1.

01:43 And then we get equal to the square root.

01:45 Now because they are independent therefore equal to the sigma square of the r plus the sigma square of the e1.

01:54 They get equal to the square root of.

01:57 Now sigma square of the r, it will equal to the two square plus sigma square.

02:02 Square of the e1 equal to the 10 square...

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