Question

true or false? Give reason
If correlation coefficient between x and y is 0.62 , then correlation coefficient
between u and v will be 0.62 , where u = 5 + 6x and v = 7 − 3y .

Answer 1

Answer on Question#65569 – Math – Statistics and Probability

Question. True or false? Give reason.

If correlation coefficient between $x$ and $y$ is 0.62, then correlation coefficient between $u$ and $v$ will be 0.62, where $u = 5 + 6x$ and $v = 7 - 3y$.

Solution. Let us start from definition of correlation coefficient

(see https://en.wikipedia.org/wiki/Pearson_corrlation_coefficient):

$$\rho_{x,y} = \frac{cov(x,y)}{\sigma_x \sigma_y}. \text{ Then } \rho_{u,v} = \frac{cov(u,v)}{\sigma_u \sigma_v}. \text{ Since } \begin{cases} u = 5 + 6x \\ v = 7 - 3y \end{cases}, \text{ using the properties of variance}$$

(see https://en.wikipedia.org/wiki/Variance#Properties) we get:

$$Var(u) = Var(5 + 6x) = 36Var(x); \quad Var(v) = Var(7 - 3y) = 9Var(y). \text{ Then}$$ $$\sigma_u = \sqrt{Var(u)} = \sqrt{36Var(x)} = 6\sqrt{Var(x)} = 6\sigma_x. \text{ Similarly } \sigma_v = 3\sigma_y.$$ $$cov(x,y) = E\left[(x - E(x))(y - E(y))\right]$$

(see https://en.wikipedia.org/wiki/Pearson_corrlation_coefficient). Then

$$cov(u,v) = E\left[(u - E(u))(v - E(v))\right]. \text{ Using the properties of mathematical expectation}$$

(see https://en.wikipedia.org/wiki/Expected_value#Properties) we get:

$$u - E(u) = 5 + 6x - E(5 + 6x) = 5 + 6x - (5 + 6E(x)) = 6(x - E(x)). \text{ Similarly}$$ $$v - E(v) = 7 - 3y - E(7 - 3y) = 7 - 3y - (7 - 3E(y)) = -3(y - E(y)). \text{ Then we have:}$$ $$\rho_{u,v} = \frac{E\left[6(x - E(x)) \cdot (-3)(y - E(y))\right]}{6\sigma_x \cdot 3\sigma_y} = \frac{-18E\left[(x - E(x))(y - E(y))\right]}{18\sigma_x \sigma_y} = -\rho_{x,y} = -0.62 \neq 0.62.$$

The statement is false.

Answer. False.

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