Expectation of terms involving indicator function

Question

Let $Z \sim \mathcal{N}(0,1)$ and $\textbf{1}\{Z > c\}$ be the indicator function such that it takes value of $1$ when $Z > c$ and $0$ otherwise. Find the expectation $E[Z\cdot\textbf{1}\{Z > c\}]$ and the covariance of the two.

Answer 1

$E[Z1{Z>c}]=\int_c^{\infty} x\phi (x)dx$ where $\phi (x)=\frac 1 {\sqrt {2\pi}} e^{-x^{2}/2}$. Since the integrand is the derivative of $-\frac 1 {\sqrt {2\pi}} e^{-x^{2}/2}$ the value of $E[Z1{Z>c}]$ is $\frac 1 {\sqrt {2\pi}} e^{-c^{2}/2}$.

Answer 2

In general:$$\mathbb EZ\mathbf1_{Z>c}=\int_{-\infty}^{\infty}z\mathbf1_{z>c}dF_Z(z)=\int_c^{\infty}zdF_Z(z)$$where $F_Z$ denotes the CDF of $Z$ and $\mathbf1_{z>c}$ is a notation for $\mathbf1_{(c,\infty)}(z)$.

If $Z$ has a PDF $f_Z$ then:$$\cdots=\int_c^{\infty}zf_Z(z)dz$$ So in your case:$$\int_c^{\infty}z\phi(z)dz$$where $\phi$ denotes the PDF of a standard normal distribution.