Expectation involving multiple RV's

$X$ follows $N(0,1).$ I am supposed to find $E[X\phi(X)]$, where $\phi(X)$ is the CDF of X

I know that $X$~ $N(0,1)$ and $Y = \phi(X)$~$U[0,1]$ , but I am not able to find the distribution of $XY$ because they are not independent.

How do I solve this?

Solution 1:

The CDF of X is $\Phi(x)$; $\phi(x)$ is its density. This is important because when you calculate the requested mean with L.O.T.U.S. you have

$$\mathbb{E}[X\Phi_X(x)]=\int_{-\infty}^{\infty}x\Phi(x)\phi(x)dx$$

and you can solve it by parts

$$\begin{align} \mathbb{E}[X\Phi_X(x)] & =\int_{-\infty}^{\infty}\underbrace{\Phi(x)}_{f}\cdot \underbrace{x\phi(x)}_{g'}dx\\ &=0+\int_{-\infty}^{\infty}\frac{1}{2\pi}e^{-x^2/2}e^{-x^2/2}dx\\ & =\frac{1}{2\pi}\sqrt{\pi}\\ & =\frac{1}{2\sqrt{\pi}} \end{align}$$