Question

Which of the following statements are true or false? Give reasons for your answers. iv) If correlation coefficient between x and y is 0.62., then correlation coefficient between u and v will by 0.62, where U=5+6xand V =7-3y

Answer 1

Answer on Question#65582 – Math – Statistics and Probability

Question. Which of the following statements are true or false? Give reason for your answers.

iv) 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. We shall use the following formulas:

$$\rho_{x,y} = \frac{cov(x,y)}{\sigma_x\sigma_y} = \frac{E(xy) - E(x)E(y)}{\sigma_x\sigma_y}$$

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

$\rho_{u,v} = \frac{E(uv) - E(u)E(v)}{\sigma_u\sigma_v}$. During computation $\rho_{u,v}$ we shall use the properties of mathematical expectation and standard deviation

(see https://en.wikipedia.org/wiki/Expected_value#Linearity

and

https://en.wikipedia.org/wiki/Standard_deviation#Identities_and_mathematical_properties).

So

$$\rho_{u,v} = \frac{E(uv) - E(u)E(v)}{\sigma_u\sigma_v} = \frac{E[(5 + 6x)(7 - 3y)] - E(5 + 6x) \cdot E(7 - 3y)}{\sigma(5 + 6x) \cdot \sigma(7 - 3y)} = \frac{E(35 - 15y + 42x - 18xy) - (5 + 6E(x))(7 - 3E(y))}{6\sigma_x \cdot 3\sigma_y}$$ $$= \frac{35 - 15E(y) + 42E(x) - 18E(xy) - 35 + 15E(y) - 42E(x) + 18E(x)E(y)}{18\sigma_x\sigma_y} = \frac{-18(E(xy) - E(x)E(y))}{18\sigma_x\sigma_y} = -\frac{E(xy) - E(x)E(y)}{\sigma_x\sigma_y}$$ $$= -\rho_{x,y} = -0.62 \neq 0.62, \text{ and we conclude that the statement is false}.$$

Answer. False.

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