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Suppose we have $X_1, X_2,\dots X_n$ from some distribution $F$. We can then form the empirical distribution $F_n$ by saying:

$$F_n(x)=\frac{1}{n}\sum_{i=1}^n\mathbb{I}_{\{X_n\leq x\}}$$

I find it hard to visualize why does this distribution put weight $\frac{1}{n}$ to each observation $X_i$ since the graph is a step function. Any help is appreciated

asdf
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    What are the sizes of the steps? – whuber May 27 '18 at 22:09
  • They are indeed $\frac{1}{n}$, but every step interval has length $X_{(i+1)}-X_{(i)}$, shouldn't this affect the weight? – asdf May 28 '18 at 08:11
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    The "weights" are merely the magnitudes of the jumps. The connection is formally made in my answer at https://stats.stackexchange.com/a/73626/919. – whuber May 28 '18 at 12:41
  • I wrote a blog post about it. I explain what the Dirac-delta function is and how it can be used for sampling, [here](https://maurocamaraescudero.netlify.app/post/towards-smc-using-the-dirac-delta-function-in-sampling-and-sequential-monte-carlo/) – Euler_Salter May 19 '20 at 15:39

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