I'm having this counterintuitive result that hopefully you fellas can shed me some light.
Imagine a discrete p.d.f with n classes. The mean of this distribution can be written as:
$$ E[X] = \sum_{i=1}^n w_ix_i $$
I can ask the question: what would happen if I increase the probability of a certain class, say class $j$. A way to imagine is that is to take the derivative:
$$ \partial_{w_j} E[X]= \partial_{w_j}w_1x_1+\partial_{w_j}w_2x_2+...+\partial_{w_j}w_jx_j+...+\partial_{w_j}w_nx_n $$ $$\partial_{w_j} E[X] = \partial_{w_j}w_jx_j = x_j $$
So far, so good, but this result seems counterintuitive when you think of in terms of a distribution. Imagine that all $x_i$ are positive. That would imply the mean should always increase even if I'm increasing the probability of the smallest value. How can this be correct?
I tried to correct by assuming trying to add the constrain that $\sum_iw_i=1$,but that essentially becomes cyclical. I can write $w_i(w_j(w_i(w_j.....$
Any ideas how to estimate the effect of increasing the probability of one class on the mean?