Question

Suppose X and Y are independent r.v and their S.D is 3 and 5 then V(Y - X) =
(a) 2
(b) 16
(c) 31
(d) 8
(e) none of the above​

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

Answer:

Step-by-step explanation:

a) Let X and Y be random variables with

E( X ) = µ X = 20, SD( X ) = σ X = 7,

E( Y ) = µ Y = 8, SD( Y ) = σ Y = 2, Corr( X, Y ) = ρ = 0.60.

Find E( 3 X – 5 Y ) and SD( 3 X – 5 Y ).

E( 3 X – 5 Y ) = 3 µ X – 5 µ Y = 3 ⋅ 20 – 5 ⋅ 8 = 20.

Var( 3 X – 5 Y ) = Cov( 3 X – 5 Y, 3 X – 5 Y )

= Cov ( 3 X, 3 X ) – Cov( 3 X, 5 Y ) – Cov( 5 Y, 3 X ) + Cov ( 5 Y, 5 Y )

= 9 2 σ X – 30 σ XY + 25 2 σ Y = 9 2 σ X – 30 ρ σ X σ Y + 25 2 σ Y

= 9 ⋅ 72 – 30 ⋅ 0.6 ⋅ 7 ⋅ 2 + 25 ⋅ 22 = 289.

SD( 3 X – 5 Y ) = 289 = 17.

b) Let X and Y be random variables with

SD( X ) = σ X = 5, SD( Y ) = σ Y = 7, SD( 3 X + 4 Y ) = 27.

Find Corr( X, Y ) = ρ.

Var( 3 X + 4 Y ) = Cov( 3 X + 4 Y, 3 X + 4 Y )

= Cov ( 3 X, 3 X ) + Cov( 3 X, 4 Y ) + Cov ( 4 Y, 3 X ) + Cov ( 4 Y, 4 Y )

= 9 2 σ X + 24 σ XY + 16 2 σ Y = 9 2 σ X + 24 ρ σ X σ Y + 16 2 σ Y

= 9 ⋅ 52 + 24 ⋅ ρ⋅ 5 ⋅ 7 + 16 ⋅ 72 = 1009 + 840 ⋅ ρ = 729 = 27 2.

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