Let X be a square-integrable a random variables. Starting from the projection definition of aconditional 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.

Given that X be a square-integrable random variables.

Answered: Let X be a square-integrable a random…

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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This result leads immediately to an important generalization. Consider a function of X and Y in the form g(X)h(Y ) for which an expectation exists. Then, if X and Y are independent: E[g(X)h(Y)]=E[g(X)JE[h(Y)]
Let X have pdf { [ kr(1-2), 0
Let f(x) = Mean Value Theorem. x² and g(x) = xf(x). Find c € (0, 1) as guaranteed by the Cauchy
1. Show (in terms of e-6) that a function f: [2, 7] R be defined by f(r) = Vr + I is uniformly continuous. %3D
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Theorem 6.4 Let f(x) 2 0 and g(x) 2 0 be two measurable functions on the set E. If h(x) = f(x) + g(x), then Sh(x)dx = J f(x)dx + J g(x)dx
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Let X and Y be random variables with joint den- √(x+2y) x,y>0,x+y Y). (b) Calculate fx y(xly). sity: f(x,y): = (c) Calculate P(X < 1 Y = 1). (d) Calculate E[X²|Y].
If X and Y are continuous random variables such that Y|X over [x, x + 1] and X is uniformly distributed over [0, 1], find the joint pdf of X and Y (including its domain), P[X + Y < 1], and E[X|Y]. = x is uniformly distributed
Let X be a strictly positive r.v. with the following moment generating function:MX (t) = 1/(1 − 2t)^3(a) Use Markov’s inequality to find an upper bound for P (|X| ≥ 18).(b) Use Chebyshev’s inequality to find an upper bound for P (|X − 6| ≥ 12).(c) Find the lowest possible upper bound for P (X ≥ 18) you could get fromChernoff’s inequality.
Assume that S={O1, O2, O3, O4} and that w1 = 0.22, w2 = 0.12, w3 = 0.17, w4 = 0.49. Let E={O2, O3} and F={O3, O4}. (a) What is the probability Pr[E]? (b) What is Pr[F′]? (c) What is Pr[(E∪F)′]?

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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]{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.