# Standard Normal Distribution with Conditional Expectation

Consider the random variable $$Z$$ that has a Normal distribution with mean $$0$$ and variance $$1$$, i.e $$Z\sim{N(0,1)}$$. I have to show that the expectation of $$Z$$ given that $$a is given by $$\begin{equation} \frac{\phi(a)-\phi(b)}{\Phi(b)-\Phi(a)} \end{equation}$$ where $$\Phi$$ denotes the cumulative distribution function for $$Z$$.

I attempted to compute first $$P(Z|a by writing it as a fraction, i.e $$\frac{P(Z=z \cap a The denominator is pretty straight forward but I cannot really get anywhere with the numerator.

• Do you need further explanation?
– NCh
Jan 26 '20 at 7:43

Use $$\mathbb E[Z|A]=\frac{\mathbb E[Z\mathbb 1_{A}]}{\mathbb P(A)}$$. Here $$\mathbb E[Z|a The numerator can be found directly: $$\int\limits_a^b z\phi(z)\,dz = \int\limits_a^b z\frac{1}{\sqrt{2\pi}}e^{-z^2/2}\,dz = \frac{1}{\sqrt{2\pi}}\int\limits_a^b e^{-z^2/2}\,d(z^2/2) = -\frac{1}{\sqrt{2\pi}}e^{-z^2/2}\biggm|_a^b$$ $$= \phi(a)-\phi(b).$$