a) Suppose that, for -1 < a < 1, the probability density function of (X₁, X₂) is given by f(x₁, x₂) = {11 — α(1 – 2e-*¹)(1 – 2e¯×²)]e*¹-*²,0 ≤×₁,0 ≤ x₂. otherwise i) Find the marginal distribution of X₁. ii) Find E(X₁X₂).
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![a) Suppose that, for -1 < a < 1, the probability density function of (X₁, X₂) is given by
f(x₁, x₂) = {11-
S[1 - α(1 - 2e-x₁)(1 - 2e-₂)] ex₁-x2 , 0 ≤ x1, 0 S Xz.
otherwise
i) Find the marginal distribution of X₁.
ii) Find E(X₁X₂).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F898a8599-529c-4956-85cd-c4240bfacfd7%2Faff1e6fe-4581-45c7-a507-a5fff48ff0cf%2F45g52xb_processed.jpeg&w=3840&q=75)
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- Suppose that X and Y have a joint probability density function f(x,y)= 1, if0<y<1,y<x<2y; 0, otherwise. (a) Compute P(X + Y less than or equal 1). (b) Find the marginal probability density functions for X and Y , respectively. (c) Are X and Y independent?Suppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise c=1/2 P(X < 1), Determine whether X and Y are independentSuppose the joint probability density of X and Y is fX,Y (x, y) = 3y 2 with 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 and zero everywhere else. 1. Compute E[X|Y = y]. 2. Compute E[X3 + X|X < .5]
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- For the probability density function f(x) = 3x^2 on [0,1], find: V(X)Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fXY(x, y), (b) fYZ(y, z), (c) fZ(z)Suppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise Find the constant c, P(Y≥1/2), P(X < 2, Y >1/2), P(X < 1), Determine whether X and Y are independent.