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I first came across this equation $\mathbb{E}_{x\sim p(x)}f(x)$ in a GAN paper but now I have seen the same equations in many different machine learning and statistical papers. While searching I came to know that its the expectation of f(x) with respect to the distribution p for variable x ref1, ref2. Can you please tell me what is the meaning of this statement, possibly with a simple example?

PS: Although I have basic idea of linear algebra, calculus and statistics , I am not an expert in math itself.

Eka
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  • It is difficult to answer without knowing your level of knowledge. If $p(x)$ is a probability density function on the real numbers, then it is $\int\limits_{x=-\infty}^{\infty} f(x) ,p(x) , dx$ while if $p(x)$ is the probability mass function then it is $\sum\limits_{x} f(x) ,p(x)$ – Henry May 09 '21 at 09:41
  • Isn't this $\sum\limits_{x} f(x) ,p(x)$ the equation for expectation itself? Is $\mathbb{E}{x\sim p(x)}f(x)$ and $\sum\limits{x} f(x) ,p(x)$ the same? Is it possible to give the intuition/logic behind this equation? Intuition like, expected value of a distribution is the average most likely seen value. – Eka May 09 '21 at 12:17
  • In case this helps: https://math.stackexchange.com/a/4050194/312 – leonbloy May 10 '21 at 02:32

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