Mathematical expectation | Wikipedia audio article
Channel: wikipedia tts
Date: 07/10/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Expected_value


00:00:28 1 History
00:00:56 1.1 Etymology
00:01:24 2 Definition
00:01:38 2.1 Finite case
00:02:20 2.1.1 Examples
00:02:48 2.2 Countably infinite case
00:03:30 2.2.1 Examples
00:04:40 2.3 Absolutely continuous case
00:06:33 2.4 General case
00:07:01 3 Uses and applications
00:08:25 4 Basic properties
00:09:21 5 Interchanging limits and expectation
00:12:10 6 Inequalities
00:14:30 7 Relationship with characteristic function
00:16:22 8 Alternative formula for expected value
00:17:18 8.1 Finite and countably infinite case
00:17:47 8.1.1 Example
00:18:15 8.2 General case
00:18:43 9 See also



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SUMMARY
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In probability theory, the expected value of a random variable is a key aspect of its probability distribution. Intuitively, a random variable's expected value represents the average of a large number of independent realizations of the random variable. For example, the expected value of rolling a six-sided die is 3.5, because the average of all the numbers that come up converges to 3.5 as the number of rolls  approaches infinity (see § Examples for details). The expected value is also known as the expectation, mathematical expectation, mean, or first moment.
A bit more formally, the expected value of a discrete random variable is the probability-weighted average of all its possible values. In other words, each possible value the random variable can assume is multiplied by its probability of occurring, and the resulting products are summed to produce the expected value. The same principle applies to an absolutely continuous random variable, except that an integral of the variable with respect to its probability density replaces the sum. The formal definition subsumes both of these and also works for distributions which are  neither discrete nor absolutely continuous; the expected value of a random variable is the integral of the random variable with respect to its probability measure.The expectation of a random variable plays an important role in a variety of contexts. For example, in decision theory, an agent making an optimal choice in the context of incomplete information is often assumed to maximize the expected value of their  utility function.  
For a different example, in statistics, where one seeks estimates for unknown parameters based on available data, the estimate itself is a random variable. In such settings, a desirable criteria for a "good" estimator is that it is  unbiased – that is, the expected value of the estimate is equal to the true value of the underlying parameter.