Suppose random variable D has C.D.F. F. D is demand and y is supply in this case. Now, excess demand (D-y), D>y is lost and excess inventory (y-D), y>D is wasted. I have to find Expectation of lost demand, ie E(D-y) given D>y.
What I have is $$E(D-y)^{+}=\int_{y}^{\infty }xdF(x)-y(1-F(y)),$$ but I'm not able to work my way through. Detailed description will be very helpful.
Thank you.
It may be better look at this in the perspective of a conditional expectation since you want to know expected demand given that it is in excess (i.e. expected excess demand) stated as $E(D|D>y)$ now this relies on finding what $P(D\leq t|D>y)$ looks like which would be $$F_{D|D>y}=P(D\leq t|D>y)=\frac{\int_{y}^{t}f(x)dx}{\int_{y}^{\infty}f(x)dx}=$$ Thus pdf for $D|D>y$ is merely $$f_{D|D>y}(x)=\frac{1}{\int_{y}^{\infty}f(x)dx}f(x)\textrm{ for }x>y$$ Thus expected excess demand would be (i represent demand by x instead of d to avoid confusion) $$E(D|D>y)=\frac{1}{\int_{y}^{\infty}f(x)dx}\int_{y}^{\infty}xf(x)dx$$
Note: I worked this up kind of late so if anyone has any corrections or comments please do so