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For a different 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. It is possible to construct an expected value equal to the probability of an event by taking the expectation of an indicator function that is one if ...
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of ...
For example, while a given person has a specific age, height and weight, ... The covariance matrix is the expected value, element by element, of the ...
The proposition in probability theory known as the law of total expectation, [1] the law of iterated expectations [2] (LIE), Adam's law, [3] the tower rule, [4] and the smoothing theorem, [5] among other names, states that if is a random variable whose expected value is defined, and is any random variable on the same probability space, then
If p = 1/n and X is geometrically distributed with parameter p, then the distribution of X/n approaches an exponential distribution with expected value 1 as n → ∞, since (/ >) = (>) = = = [()] [] =. More generally, if p = λ/n, where λ is a parameter, then as n→ ∞ the distribution of X/n approaches an exponential distribution with rate ...
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. Stopped Brownian motion is an example of a martingale. It can model an even coin-toss ...
In cases where (,) are such that the conditional expected value is linear; that is, in cases where = +, it follows from the bilinearity of covariance that = (,) and = (,) and the explained component of the variance divided by the total variance is just the square of the correlation between and ; that is, in such ...
Cost := Value_per_minute_at_home * Time_I_leave_home + (If Time_I_leave_home < Time_from_home_to_gate Then Loss_if_miss_the_plane Else 0) The following graph displays the expected value taking uncertainty into account (the smooth blue curve) to the expected utility ignoring uncertainty, graphed as a function of the decision variable.