Let X be a square-integrable a random variables. Starting from the projection definition of a conditional expectation, prove that E[X]{Ø, S}] = E[X], E[X]o(X)] = X, and E[g(X)o(X)] = g(X), where g is a Borel function and E[g²(X)] < x.
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![Let X be a square-integrable a random variables. Starting from the projection definition of a
conditional expectation, prove that E[X]{0, 2}] = E[X], E[X]o(X)] = X, and E[g(X)|o(X)] = g(X),
where g is a Borel function and E[g²(X)] < x.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71f6a4b8-527f-4030-b85a-828002701c6f%2F6f4f0e76-2552-4a9d-addf-6606adb42baa%2Fy7uqecr_processed.png&w=3840&q=75)
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- Let X be a random variable on a closed and bounded interval [a, b]. Let g(x) be a convex function. Prove that g(E(X)) ≤ E (g(X)Let f(x) = Mean Value Theorem. x² and g(x) = xf(x). Find c € (0, 1) as guaranteed by the CauchyLet a pdf of random vector X be given as 3 where a = [a₁, fx(x) = K · exp(-x¹a)u(x) an] T. Find the value of K. 7
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