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In the Case of a Simple Regression, Where the Independent

question 44

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In the case of a simple regression, where the independent variable is measured with i.i.d. error, a. β^1pσX2σX2+σw2β1\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 }

b. β^1pσX2σX2+σw2\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } .

c. β^1pσw2σX2+σw2β1\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \frac { \sigma _ { w } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 } .

d. β^1pβ1+σX2σX2+σw2\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \beta _ { 1 } + \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } .

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Definitions:

Probability

The measure of the likelihood that an event will occur, often represented as a number between 0 and 1.

Negation

The logical operation that inverts the truth value of a proposition or statement.

Probability

The degree to which something is probable; the extent to which an event is likely to happen.

Event A

A specific occurrence or happening, often denoted in probability and statistics as a measurable outcome.

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