My question is: How do I calculate the partial expected value of the part of a normal distribution that ranges from infinity (at the left side) to a number Before or less than the mean. Or of coourse symmetrically the corresponding range at the right of the distribution. I basically wonder how to 'split' the distribution in three parts.
Greatful for answers
Your question is tagged with integration and calculus, so I'll answer with those. However, it's not really clear to me what exactly you want to know, so let me know if this is helpful.
Assuming $X\sim \mathcal{N}(\mu,\sigma^2)$ is normally distributed with density function $f(x|\mu,\sigma^2)$ and $a \in\mathbb{R}$ is some fixed number smaller than the mean, then $$ P(X < \mu-a) = \int_{-\infty}^{\mu-a} f(x|\mu,\sigma^2) \,dx $$ The other "two parts" are $$ P(\mu-a < X < \mu+a) = \int_{\mu-a}^{\mu+a} f(x|\mu,\sigma^2) \,dx $$ $$ P(X > \mu+a) = \int^{\infty}_{\mu+a} f(x|\mu,\sigma^2) \,dx $$ If you want to consider these in terms of the standard Gaussian CDF $\Phi$, notice that: $$ P(X < \mu-a) = P(\sigma Z + \mu < \mu - a)=P(Z<-a/\sigma) = \Phi\left( \frac{-a}{\sigma} \right) $$ where $Z = (X-\mu)/\sigma$.
To find the "middle segment", for instance, you can see that $$ P(\mu-a < X < \mu+a) = P(-a/\sigma < Z < a/\sigma) = \Phi\left(\frac{a}{\sigma}\right) - \Phi\left(\frac{-a}{\sigma}\right) $$ With these, you can look up values in tables like these (use the cumulative one) or do the integration directly.