# Dirac Delta function Notation

I am trying to understand the delta function notation used to be express a monte carlo approximation of a probability distribution. The notation used in this (p10) is

$$\pi(x_{1:n}) = \frac{1}{N}\sum^N_{i=1}\delta_{X^i_{1:n}}(x_{1:n})$$

where $$X^i_{1:n} \sim \pi(x_{1:n})$$ for i=1,..,N

I'm wondering whether this is equivalent to the following

$$\pi(x_{1:n}) = \frac{1}{N}\sum^N_{i=1}\delta(x_{1:n}-{X^i_{1:n}})$$

for a specific value of $$x_{1:n}$$, say $$x_{1:n} = y$$. I understand the latter to mean that the probability of a particular $$x_{1:n}$$ is given by the number of samples that take this value, (hence $$\delta(y - X^i_{1:n}) = \delta(0) = 1 )$$ divided by the total number of samples, N.

Is this correct?

I realise my questions is similar to the question asked here: Delta Function in Monte Carlo Sampling and have read it and looked at linked answers.

• – whuber
Commented Apr 5, 2016 at 21:38
• This Monte Carlo approximation consists in replacing the unknown distribution $\pi$ with a finite support distribution, which support is the simulated sample $X_{1:n}$. This is also the basis of bootstrap. Commented Apr 8, 2016 at 17:04
• I wrote a blog post about it. I explain what the Dirac-delta function is and how it can be used for sampling, here Commented May 19, 2020 at 15:38