Show That the Correlation Coefficient Between Y and $\operatorname { corr } ( X , Y ) = \operatorname { corr } \left( X ^ { * } , Y ^ { * } \right) \text {, }$

question 10

Essay

Show that the correlation coefficient between Y and X is unaffected if you use a linear transformation in both variables. That is, show that $\operatorname { corr } ( X , Y ) = \operatorname { corr } \left( X ^ { * } , Y ^ { * } \right) \text {, }$ where $X ^ { * } = a + b X \text { and } Y ^ { * } = c + d Y \text {, }$ and where a, b, c , and d are arbitrary non-zero constants.

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Definitions:

Marginal Cost Curve

A graph showing the change in the cost of producing one more unit of a good as production levels change.

Short-Run Supply Curve

A graphical representation showing the quantity of goods a firm is willing to supply at different price levels in the short term, holding some inputs fixed.

Shutdown Point

The level of operation at which a firm's total revenue is equal to its total variable costs, below which the firm should cease production to minimize losses.

Point

A specific location or position in a spatial setting or within a set of data.