
Expectation
Hey,
I have given a normally distributed random variable $\displaystyle $X\stackrel{d}{=}\mathcal{N}(\mu,\sigma^2)$$ and want to compute the following expectation
$\displaystyle $\mathbb{E}[X \mathbb{1}_{\{X\geq 1\}}]$$
where $\displaystyle $\mathbb{1}$$ denotes the indicator function.
I think that it is not possible to compute the corresponding integral explicitly. Am I right?
Thanks in advance!

Re: Expectation
Hey Juju.
Can you find the random variable corresponding to X >= 1 where X has that Normal distribution?
Hint: Use the conditional distribution of XX>=1 where P(AB) = P(A and B)/P(B).

Re: Expectation
Thank you for your reply,
unfortunately I do not understand what you mean by "corresponding random variable". Of course I can determine the distribution function of $\displaystyle X \mathds{1}_{\{X\geq 1\}}$ and the expectation is given by
$\displaystyle \int\limits_1^\infty (2 \pi \sigma^2)^{\frac{1}{2}}xe^{\frac{1}{2}\frac{(x\mu)^2}{\sigma^2}}dx$
But I cannot give this integral in a closed form.

Re: Expectation
You should be able to evaluate the integral: (Hint: use a substitution v = (xmu)^2 and change from dx to dv).
Also when I say corresponding random variable I just mean the random variable that is represented algebraically with respect to the limits of the integrand.
For example I can have Y = X + 3 and from this specify a conditional probability of P(YX) if I know the density function for the random variable X.
Now similarly, you can have limits in the integration that correspond with a particular algebraic expression but they may not have an easy interpretation algebraically (i.e. might be a lot more complex than the relation Y = X + 3).