I am reading All of Statistics from Casella. When trying to show the bias for a histogram estimator for some density distribution, he starts developing the formula for pj, that is, the probability some observations lies in that bin. Then he uses taylor approximation for that formula. However, developing the integral for that pj by myself do not show the same results from that book. Treating x as a constant, makes it to vanish. Where the j comes from? Can you develop step by step that integral?
Let's take a closer look at the bias-variance tradeoff using equation (20.9). Consider some $x \in B_j$. For any other $u \in B_j$, $$ f(u) \approx f(x)+(u-x) f^{\prime}(x) $$ and so $$ \begin{aligned} p_j=\int_{B_j} f(u) d u & \approx \int_{B_j}\left(f(x)+(u-x) f^{\prime}(x)\right) d u \\ &=f(x) h+h f^{\prime}(x)\left(h\left(j-\frac{1}{2}\right)-x\right) . \end{aligned} $$
