- #1
Steve Zissou
- 53
- 0
- TL;DR Summary
- One way to think of the Dirac Delta "function" is the limit of a normal distribution as its standard deviation collapses to an infinitesimal. What if we start with a lognormal?
Hello shipmates,
Instead of imagining a Dirac Delta as the limit of a normal, like this:
$$ \delta\left ( x \right ) = \lim_{a \to 0}\frac{1}{|a|\sqrt{2\pi}}\exp\left [ -\left ( x/a \right )^2 \right ] $$
Could we say the same thing except starting with a lognormal, like this?
$$ \delta_{LN} \left ( x \right ) = \lim_{a \to 0}\frac{1}{|a|x\sqrt{2\pi}}\exp\left [ -\left ( \log{x}/a \right )^2 \right ] $$
Thanks!
Your pal,
Stevsie
Instead of imagining a Dirac Delta as the limit of a normal, like this:
$$ \delta\left ( x \right ) = \lim_{a \to 0}\frac{1}{|a|\sqrt{2\pi}}\exp\left [ -\left ( x/a \right )^2 \right ] $$
Could we say the same thing except starting with a lognormal, like this?
$$ \delta_{LN} \left ( x \right ) = \lim_{a \to 0}\frac{1}{|a|x\sqrt{2\pi}}\exp\left [ -\left ( \log{x}/a \right )^2 \right ] $$
Thanks!
Your pal,
Stevsie