Correlation coefficients of X and Y

Question

I was wandering if anybody could help me with the following question. I am fairly new to correlation coefficients and was attempting to tackle this question but was unsure how to do so? Thanks.

Answer 1

a)

$Y^{2}=X^{2}$ so that $\mathbb{E}Y^{2}=\mathbb{E}X^{2}=1$.

$X$ is symmetric and consequently $Y$ is symmetric. That implies $\mathbb{E}Y=0$ so that $\text{Var}Y=\mathbb EY^2=1$.

Then: $$\rho\left(X,Y\right):=\mathbb{E}\left(\frac{X-\mu_{X}}{\sigma_{X}}\right)\left(\frac{Y-\mu_{Y}}{\sigma_{Y}}\right)=\mathbb{E}XY$$ Now work this out.

b) If $a$ approaches $0$ then the correlation approaches $-1$. If $a$ approaches $\infty$ then the correlation approaches $1$. The correlation can looked at as continuous function having $a$ as argument. With intermediate value theorem it can be proved that it takes value $0$ for some $a$ (uncorrelated).

c) Let it be that for $a_0>0$ the correlation is $0$. If $(X,Y)$ would have a bivariate normal distribution for this $a_0$ then it would legal to conclude that $X$ and $Y$ are independent in this case (characteristic for (bivariate) normal distribution is that being uncorrelated is the same as being independent). However, it is evident that $X,Y$ are not independent.