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Expectation and Variance. The expected value (or mean) of X, where X is a discrete random variable, is a weighted average of the possible values that X can take, each value being weighted according to the probability of that event occurring. The expected value of X is usually written as E (X) or m.
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average.
In probability and statistics, the expected value is the theoretical mean (this assumes that the experiment is run a relatively large number of times) of a random variable, X. For example, the experiment of rolling a fair six-sided die has six possible outcomes, all of which have an equal probability of occurring:
Essentially, if an experiment (like a game of chance) were repeated, the expected value tells us the average result we’d see in the long run. Statisticians denote it as E (X), where E is “expected value,” and X is the random variable.
The standard deviation of \(X\) is the square root of this sum: \(\sigma = \sqrt{1.05} \approx 1.0247\) The mean, μ, of a discrete probability function is the expected value. \[μ=∑(x∙P(x))\nonumber\] The standard deviation, Σ, of the PDF is the square root of the variance. \[σ=\sqrt{∑[(x – μ)2 ∙ P(x)]}\nonumber\]
Summary of the properties of the expected value operator, with explanations, proofs, examples and solved exercises. Stat Lect Index > Fundamentals of probability
Oct 20, 2018 · Consider a square-integrable random variable X X living on (Ω,F,P) (Ω, F, P). The defining property of the conditional expectation of X X w.r.t. a σ σ -field G ⊂F G ⊂ F (X~ =E[X ∣ G] X ~ = E [X ∣ G]) can be writen as E[(X −X~)Z] = 0 E [(X − X ~) Z] = 0 for all bounded, G G -measurable Z Z.
