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The textbook of probability said:

A sequence of random variables $X_1, X_2, X_3, \ldots$ converges in distribution to a random variable $X$, shown by $X_n \overset{d}{\to} X$, if $$ \lim_{n\to\infty}F_{X_n}(x) = F_X(x), $$ for all $x$ at which $F_X(x)$ is continuous.

However, my professor simply wrote it like the following: Convergence

  1. in distribution: PDF of $X_n$ $\to$ PDF of $X$
  2. in probability: ...
  3. in mean-square: ...
  4. with probability 1 (almost sure): ...

Convergence of CDF and that of PDF are same?

The reason why I ask this question is I've heard one day that $$ E(X) = \int x~ dF_X(x) $$ is true, but $$ E(X) = \int xf_X(x)~dx $$ is wrong.

Danny_Kim
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    Not every random variable has a pdf. But if it does, the the second equation is true – user251257 Oct 02 '16 at 17:47
  • @user251257 You mean, the reason why the textbooks stick to using CDF instead of PDF is special case when the random variable does not have PDF, right??? Thank you – Danny_Kim Oct 02 '16 at 19:29
  • also, characterizing convergence in distribution by PDF is much more cumbersome than using CDF. – user251257 Oct 02 '16 at 19:32

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