Question

If X is a normal random variable with mean μ and variance σ2, define a random variable Y that has the same distribution as X and is negatively correlated with it.

Short answer

Random variable$Y$ is defined as$Y=\mu -\left(X-\mu \right)=2\mu -X$ and it is explained below.

Step-by-step solution

given Information

We need to define a random variable $Y$that has the same distribution as $X$and is negatively correlated with it.

Simplify

Considering random variable $X$to be symmetric to the $X$about the mean $E\left(X\right)=\mu$. Observe that every value $X$can be written as

$X=\mu +\left(X-\mu \right)$

i.e. the sum of the mean and the distance from the mean. Define $Y$using $X$in this way

$Y:=\mu -\left(X-\mu \right)=2\mu -X$

$Y$is in fact random variable that is essential the same as $X$, but the sign of the distance is switched. Now, $Y$is normally distributed since $X$is normally distributed and we have that

$\begin{array}{c}E\left(Y\right)=E\left(2\mu -X\right)=2\mu -E\left(X\right)=\mu \\ \text{Var}\left(Y\right)=\text{Var}\left(2\mu -X\right)=\text{Var}\left(-X\right)=\text{Var}\left(X\right)={\sigma }^2\end{array}$

So, we have$Y$is equally distributed as $X$. Also, we have

$\text{Cov}\left(X,Y\right)=\text{Cov}\left(X,2\mu -X\right)=0-\text{Var}\left(X\right)=-{\sigma }^2$

So, we have that$X$and $Y$are negatively correlated.