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  3. suppose that the random variables, ξ, η have

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

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Statistics

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

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

a) Find the marginal densities f(x) =∫ 2/9 dy (limits, -1 to1-x) and f(y) =∫ 2/9 dx

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

independent.

b) Verify that μξ = ∫ x f(x) dx = 0 and μη= ∫ 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) ∫ ydy (y= -1 to y = 1-x)

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