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EXPECTED VALUE
The expected value of a random variable can be defined as the weighted average of all possible values that this random variable can take on and the weights that are used in computing this average correspond to the probability mass function in the case of a discrete random variable or probability density function in case of a continuous random variable. From a theoretical standpoint, it is the Lebesgue integral of the random variable with respect to its probability measure.
It may be intuitively understood by the Law of large numbers and the expected value when it exists, it limits the sample size and it grows to infinity. More informally it can be the long-run average of the results of many independent repetitions of an experiment and the value may not be an expectation in the ordinary sense. The "expected value" itself maybe even impossible just like the sample mean.
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In general, if X is a random variable defined on a probability space (Ω, Σ, P), then the expected value of X which is denoted by E[X] or E[X] is defined as the Lebesgue integration as follows
When this integral exists, it is defined as the expectation of X and not all random variables have a finite expected value as the integral does not converge. Two variables with identical probability distribution will have the same expected value if it is defined and it follows directly from the discrete case definition that if X is a constant random variable i.e. X = b for some real number b that is fixed then the expected value of X is also b.
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The function of X is arbitrary which is denoted as g(X) with respect to the probability density function f(x) and it is given by the inner product of f and g as follows
This is the Law of the unconscious statistician using the representations as Riemann–Stieltjes integral and Integration by parts where the formula can be restated as follows
In some cases, α is a positive real number which will be as follows
In particular, if α = 1 and Pr[X ≥ 0] = 1, then this reduces to as represented below
F symbolizes the cumulative distribution function of X and this last identity is an instance of a non-probabilistic setting known as the Layer cake representation.
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