Suppose that the random variables, ξ, η have joint uniform density f(x, y) = 2/9 in...

Suppose that the random variables, ξ, η have joint uniform density f(x, y) = 2/9

in the triangular region bounded by the lines x = -1 , y - -1 and y = 1 - x.

a) Find the marginal densities f(x) =∫ 2/9 dy (limits, -1 to 1-x) and f(y) =∫ 2/9 dx

(limits -1 to 1-y). Also show that f(x) f(y) ≠ f(x, y) so that ξ and η are not

independent.

b) Verify that μ_ξ = ∫ x f(x) dx = 0 and μ_η = ∫ y f(y) dy = 0

c) Find Var (ξ) = ∫ x² f(x) dx, Var (ζ) = ∫ y² f(y) dy and

Cov (ξ, η) = 2/9 ∫ x [ ∫ y dy ] dx (y limits -1 to 1-x, then x = -1 to x = 2)

d) Find ρ and regression curve E[η│ξ = x] = [1/f(x)] (2/9) ∫ y dy (y= -1 to y = 1-x)

Expert Solution

orchestra answered 1 year ago

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