Let X be a random variable taking three values: P(X = a₁) = P₁, P(X=a₂) = P2₁ P(X=03) = P3, where p₁ + P2 + P3 = 1 and P₁, P2, P3 € (0, 1). Let A = {X = a₁} and G = {2, 0, A, Ac}. Prove that E (X³|G) = a1₁ + 0²0² +0²P³ 14². P2 + P3

Q: Let X, Y be independent Uniform (0, 2) random variables. Define Y, if X }. Z = Compute E[Z] and…

A: X,Y ~ Uniform(0,2) E(X) = E(Y) = 0+22=1 Var(X) = Var(Y) = (2-0)212=412=13 Z = Yif X <122Yif X≥12…

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Q: Let x and y be two independent N(0, 1) random variables and Z = 1+ X + XY², W = 1+ X. Find Coy(7,W).

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Q: Let X and Y be random variables with finite variance. Prove that |ρ(X, Y)| = 1 implies that there…

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Q: (A) ơy = 0xı + Ox2 Vox1 + Ox2 +o? = Ox1 +0x2 (B ) συ .2 (C) oy = -2 x2 X1 .2 -2 (D) Oy/y = X2

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Q: If X is a random variable, prove that Cov(X,X) = σX².

A: Let X is a random variable.

Q: 3. If X and Y are independent uniform (0, 1) random variables, show that 2 for a > 0 (a + 1)(a + 2)…

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Q: Let X and Y be two independent N(0,1) random variables and consider Z = 3 +X+ 2XY² , W = 5 + X. Then…

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Q: Suppose that the joint CDF of the random variables X and Y is given by 0, x < -2 or y < –5

A: Given that Fx,y=0, x&lt;-2 or y&lt;-538, -2≥x&lt;2, -5≤y&lt;312, x≥2 and -5≤y&lt;312, -2≤x&lt;2 and…

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A: Hi! Thank you for the question, As per the honor code, we are allowed to answer three sub-parts at a…

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Q: Suppose that X andY are random variables with E(X) = 2 , E(Y) = 5 and E(X²) = 8, E(Y?) = 30 and…

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Q: Find the covariance between x and y

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Q: If X and Y are two independent random variables, then E(XY) = E(X) E(Y)

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Q: If X and Y have the joint probability distributionf(−1, 0) = 0, f(−1, 1) = 1 4 , f(0, 0) = 16 , f(0,…

A: The joint probability distribution of the random variables is given as follows:

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Q: Show that (a) Cov(aX, bY)= ab Cov(X,Y). (b) Cov(X+a, Y + b) = Cov(X,Y). (c) Cov(X, aX+b) = a Var(X).

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A: Given Information: f(x) = 130x2+4 x = -2, -1, 0 , 1 , 2

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Q: Suppose that the joint CDF of the random variables X and Y is given by: 0, x < -2 or y < -5

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Q: Suppose we have three independent random variables X, Y and Z where... Var(X + 2Y) = 13, Var(2Y +…

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Q: 17. Let I(W₁₁ W₂) =W₁-W₂. Compute Cov(X, Y). Are they independent? die die = {(W₁, W₂) : w₁ = 1, 2,…

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