Should I Normalize My Discretized Integral?

[pamparana]

Hello everyone,

I have an integral as follows:

\log(\int_{r}^{\inf}\frac{P(v,f)}{P(f)}dv)

Now, I want to evaluate this on the computer where this PDF represented by P(v, f) is build up using a joint histogram where each entry is a probability for a given value pair for v and f to occur together.

Now when I compute this integral I do it as follows:
\sum_{r}^{bins}\frac{P(v,f)}{P(f)}

where v and f are some appropriate values and bins are the number of bins along the appropriate axes in my joint histogram. Then after accumulation, I take the log of the final value.

Now, my question is that do I need to normalize this somehow? When discretizing such integrals does one normalize it by the range of the definite integral parameters?

Thanks,

Luca

 

[fresh_42]

You should make a graphic. What you basically do is to approximate the integral by a Riemann sum. This involves the split of the distance into equidistant sections.

 

[RPinPA]

You're missing a term.

As @fresh_42 says, you appear to be doing a Riemann sum, approximating $\int_a^b f(x) dx$ by $\sum_{i=1}^n f(x_i) \Delta x$ = $\Delta x \sum_{i=1}^n f(x_i)$ if all the bins are the same size. So you need the $\Delta x$.