conditional expectations value I need to calculate the following integral
$$\int_{\mu+c}^{\infty} y\cdot  \frac{1}{\sigma\sqrt{2\pi}}e^{(y-\mu-w)^2/2\sigma^2}dy$$ 
So essentially $y\sim N (\mu+w, \sigma^2)$ and im trying to calculate its expected value from $\mu+c$ to $\infty$
I know I will have to use integration by parts, but don't know how to proceed. 
More importantly however, when I calculate the integral, will I get an expression which includes $w$? I really need the $w$ in there.  
 A: Hint: setting $\sigma^2=1$ for convenience, we have that
$$\begin{align}
 \frac{\mathrm d}{\mathrm dy} \exp(-(y-\mu-w)^2/2) &= 
-(y-\mu-w)\exp(-(y-\mu-w)^2/2)\\
&= -y\exp(-(y-\mu-w)^2/2 \\
&\qquad + (\mu+w)\exp(-(y-\mu-w)^2/2
\end{align}$$
and so you should be able to write the integrand as  $(\mu+w)$ times 
a Gaussian density less a perfect differential,
and thus get the integral to work out to something involving $\Phi(\cdot)$ and an 
$\exp(g(w))$ term where $g(w)$ is a quadratic function of $w$.
A: The conditional expectation for a normal random variable with mean $\tau$ and standard deviation $\sigma$ can be found from this formula, which is available, for example, here: www.actuaries.org/LIBRARY/ASTIN/vol35no1/189.pdf ‎
$$ E[X | X > q ]={\tau + {{\sigma} \phi \left({{q-\tau} \over \sigma} \right) \over {1-\Phi \left( {{q-\tau} \over \sigma}\right)}  } },$$
where $\Phi$ is the CDF and $\phi$ is the PDF for the standard normal. The general formula that can be used to derive these type of expectations is given by 
$$ E[X | X > q ]={ {\int_q^\infty xf(x)dx} \over {P[X>q]}}$$
Putting these together with your parameters, we have
$$E[Y | Y > \mu+c  ]={  {{{\int_{\mu+c}^\infty {y \over {\sigma \sqrt{2\pi}}} 
e^{{-{\left  ( {y-\mu-w} \right)^2}} \over {2 \sigma^2}} dy} }\over {1-\Phi \left( {{c-w} \over \sigma}\right)}  }=\mu+w +\left[{\sigma {\phi \left( {c-w} \over \sigma \right) } \over {1-\Phi\left( {c-w} \over \sigma \right)}} \right] }$$
Finally, then, we can solve for the integral you want:
$${\int_{\mu+c}^\infty {y \over {\sigma \sqrt{2\pi}}} 
e^{{-{\left  ( {y-\mu-w} \right)^2}} \over {2 \sigma^2}} dy}=\left(\mu+w \right) \left[{1-\Phi\left( {c-w} \over \sigma \right)}  \right]+\sigma {\phi \left( {c-w} \over \sigma \right) }$$
