In the previous problem, I was working on some piecewise linear functions and doing calculus there. Having many cases is just error prone. Now I try to do thing more algebraically.
The first thing I notice is integrating delta function is simply sampling property, but it applies only if the integrating domain contains 0, that make three cases and it is just not pleasant.
Now, let 'simplify' using step functions.
$ \int\limits_{a}^{b}{\delta(x)f(x)dx} = (\theta(b) - \theta(a))f(x) $
Similarly, we also have this:
$ \int\limits_{a}^{b}{\theta(x)f(x)} = \theta(b)(F(b) - F(0)) - \theta(a)(F(a) - F(0)) $.
To check these, one just consider all 6 permutations of $ a $, $ b $ and $ 0 $.
The first thing I notice is integrating delta function is simply sampling property, but it applies only if the integrating domain contains 0, that make three cases and it is just not pleasant.
Now, let 'simplify' using step functions.
$ \int\limits_{a}^{b}{\delta(x)f(x)dx} = (\theta(b) - \theta(a))f(x) $
Similarly, we also have this:
$ \int\limits_{a}^{b}{\theta(x)f(x)} = \theta(b)(F(b) - F(0)) - \theta(a)(F(a) - F(0)) $.
To check these, one just consider all 6 permutations of $ a $, $ b $ and $ 0 $.
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