Consider a random variable X with a finite list x1, ..., xk of possible outcomes, each of which respectively has probability p1, ..., pk of occurring. The expectation of X is defined as
Since the probabilities must satisfy p1 + ⋅⋅⋅ + pk = 1, it is natural to interpret E[X] as a weighted average of the xi values, with weights given by their probabilities pi.
In the special case that all possible outcomes are equiprobable, that is, p1 = ⋅⋅⋅ = pk, the weighted average is given by the standard average. In the general case, the expected value takes into account the fact that some outcomes are more likely than others.
The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes in a fair way between two players, who have to end their game before it is properly finished. This problem had been debated for centuries. Many conflicting proposals and solutions had been suggested over the years when it was posed to Blaise Pascal by French writer and amateur mathematician Chevalier de Méré in 1654. Méré claimed that this problem couldn't be solved and that it showed just how flawed mathematics was when it came to its application to the real world. Pascal, being a mathematician, was provoked and determined to solve the problem once and for all.
He began to discuss the problem in the famous series of letters to Pierre de Fermat. Soon enough, they both independently came up with a solution. They solved the problem in different computational ways, but their results were identical because their computations were based on the same fundamental principle. The principle is that the value of a future gain should be directly proportional to the chance of getting it. This principle seemed to have come naturally to both of them. They were very pleased by the fact that they had found essentially the same solution, and this in turn made them absolutely convinced that they had solved the problem conclusively; however, they did not publish their findings. They only informed a small circle of mutual scientific friends in Paris about it.
In Dutch mathematician Christiaan Huygens' book, he considered the problem of points, and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens published his treatise in 1657 and can be seen as the first successful attempt at laying down the foundations of the theory of probability.
During his visit to France in 1655, Huygens learned about de Méré's Problem. From his correspondence with Carcavine a year later in 1656 he realized his method was essentially the same as Pascal's. Therefore, he knew about Pascal's priority in this subject before his book went to press in 1657.
In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the expectations of random variables.
Neither Pascal nor Huygens used the term "expectation" in its modern sense. More than a hundred years later, in 1814, Pierre-Simon Laplace published his tract "Théorie analytique des probabilités", where the concept of expected value was defined explicitly.
QUESTION I: How does the Expectation defined above equate to Mean of a Frequency Distribution?
QUESTION II: How does one calculate the Expectation of a continuous Random Variable?
REFERENCE: Expected Value Wikipedia