# The expected value

Simple explanations for the most common types of expected value formula. Includes video. Hundreds of statistics articles and vidoes. Free help. In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the experiment it represents. For example Definition · General definition · Properties · Uses and applications. Printer-friendly version. Expected Value (i.e., Mean) of a Discrete Random Variable. Law of Large Numbers: Given a large number of repeated trials, the average.
In the continuous case, the results are completely analogous. If one rolls the die n times and computes the average arithmetic mean of the results, then as n grows, the average will almost surely converge to the expected value, a fact known as the strong law of large numbers. The odds that you lose are out of For example, the expected value in rolling a six-sided die is 3. Given a large number of repeated trials, the average of the results will be approximately equal to the expected value. Sat Jul 8 EV can be calculated for single discreet variables, single continuous variables, multiple discreet variables and multiple continuous variables. This is utilized in covariance matrices. Interaction Help About Wikipedia Community portal Recent changes Contact page. If the outcomes x i are not equally probable, then the simple average must be replaced with the weighted average, which takes into account the fact that some outcomes are more likely than the. The third equality follows from a basic application of the Fubini—Tonelli theorem. The convergence is relatively slow: Make sizzling fruits online probability chart the expected value Collection of teaching and learning tools built by Wolfram education experts: Less technically inclined readers can safely skip it, while interested readers can read more about it in the lecture entitled Expected value and the Lebesgue integral. Y does not imply existence of E X. EV can be calculated for single discreet variables, single continuous variables, multiple discreet variables and multiple continuous variables. Probability - 1 Variable Lesson 4: Views Read Edit View history. The same principle applies to a continuous random variable , except that an integral of the variable with respect to its probability density replaces the sum. Definition, Word Problems T-Distribution Non Normal Distribution Chi Square Design of Experiments Multivariate Analysis Sampling in Statistics: This last identity is an instance of what, in a non-probabilistic setting, has been called the layer cake representation. For example, suppose X is a discrete random variable with values x i and corresponding probabilities p i. The expected value of this scenario is:. The use of the letter E to denote expected value goes back to W. They were very pleased by the fact that they had found essentially the same solution and this in turn made them absolutely convinced they had solved the problem conclusively. A More Complicated Expected Value Example The logic of EV can be used to find solutions to more complicated problems. For absolutely continuous random variables the proof is In general, the linearity property is a consequence of the transformation theorem and of the fact that the Riemann-Stieltjes integral is a linear operator:

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