Finding pearson correlation coefficient between two variables using the coefficient of one of them and a third veriable

Question

I have been given three variables: x,y and z.
y is a linear transformation of x: y=15x+20.
The frequency distribution for x is also given.
the pearson correlation coefficient between x and z is given to be r,
And I have been asked to exspress the pearson correlation coefficient between y and z.
But as far has I can tell there is no why to use any of the coefficients to infer about the other. Can it be done? What am I missing?

Answer 1

The correlation between $y$ and $z$ is the same as that between $x$ and $z.$ If you had multiplied by any negative number rather than by $15,$ then the correlation would have been multiplied by $-1.$

\begin{align} \operatorname{corr}(y,z) & = \frac{\operatorname{cov}(y,z)}{\operatorname{s.d.}(y) \operatorname{s.d.}(z)} \\[10pt] & = \frac{\operatorname{cov}(15x+20,z)}{\operatorname{s.d.}(15x+20) \operatorname{s.d.}(z)} = \frac{15\operatorname{cov}(x,z)}{|15|\operatorname{s.d.}(x)\operatorname{s.d.}(z)} \\[10pt] & = \frac{\operatorname{cov}(x,z)}{\operatorname{s.d.}(x)\operatorname{s.d.}(z)} = \operatorname{corr}(x,z). \end{align} If you had multiplied by $-15$, then you'd have $\dfrac{-15}{\left|-15\right|} = -1,$ instead of $\dfrac{15}{|15|}=1$ and you'd end up multiplying by $-1.$