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johnG2011
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Homework Statement
Let X be a continuous random variable. What value of b minimizes E(|X-b|)? Give the derivation.
This statement refers to the process of minimizing the expected value of the absolute difference between a random variable X and a constant b, where b is a continuous random variable. In other words, it involves finding the value of b that minimizes the average distance between X and b.
Minimizing E[|X-b|] can be useful in various statistical and machine learning applications. For example, it can help in reducing the error or uncertainty associated with predicting certain outcomes based on a continuous random variable. It can also help in identifying the most likely value of b that is closest to X.
E[|X-b|] is calculated by taking the expectation or average value of the absolute difference between X and b. This involves integrating the absolute difference function over the entire range of values of X.
There are various optimization techniques that can be used to minimize E[|X-b|]. These include analytical methods such as calculus, as well as numerical methods such as gradient descent or simulated annealing. The choice of technique depends on the specific problem and the constraints involved.
Yes, it is possible for E[|X-b|] to be equal to 0. This occurs when the value of b is equal to the expected value of X. In other words, when b is the same as the average value of X, there is no difference between X and b, and the absolute difference will always be 0. This is known as the minimum mean absolute error (MMAE) solution.