TITLE: What is the motivation for the definition of the expected value?
QUESTION [7 upvotes]: I have a general question about expected values: 

For a discrete random variable, $$E[X] = \sum_{i=1}^{\infty} x_{i}p_{i}$$ and $$E[X] = \int_{-\infty}^{\infty} xp(x) \ dx$$ for a continuous random variable $X$. 

But what is the motivation for these definitions? Is it essentially defined because of the following: Suppose I perform an experiment a large number of times and record the results. I then take the average of the results. I want to find what this average approaches as the number of trials increases. Thus by trial and error I find that the definition of $E(X)$ works. This is assuming I am taking a frequentist view of probability. What does this all have to do with computing the area under some function?

REPLY [2 votes]: Professor Peter Whittle, Cambridge University, argued that  average (expected value) was a more intuitive notion than probability, frequentist or not. For this reason he axiomatized expectation rather than probability. Kolmogorov axiomatized probability in 1933, and this is regarded as a milestone (millstone?) in the history of Probability Theory.
Whittle wrote a book in 1970 called Probability via Expectation. This book has been translated into Russian and other languages, and is now in its 4th Edition, 2000. Read the introductory chapter here: Probabilty via Expectation
Now, if you think that Peter Whittle's view of randomness is eccentric, then read what the great Rudolf Kalman, of Kalman Filter fame, has to say about Kolmogorov probability here:
www.scribd.com/document/62685671/Kalman-Probability-in-the-Real-World
where he answers the question:  Why is probability not a satisfactory way of looking at randomness?
Axiomatic Probability or Expectation? Take your pick -- randomly, of course.