So it is an unreasonable assumption 45 the joint pmf

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Unformatted text preview: σx = 10(25) = Var(L1 ) + Var(L2 ) + · · · + Var(L10 ). This is true only when L1 , L2 , . . . , L10 are mutually uncorrelated. So, it is an unreasonable assumption. 45. The joint pmf: x -1 0 1 y 1 0 1 p(x, y ) 1 3 1 3 1 3 The marginal pmf’s of X and Y are x px ( x ) y py ( y ) 1 -1 1 3 0 1 3 0 2 3 1 1 3 1 3 ￿ ￿￿ ￿ a. Since p(−1, 1) = 1 ￿= 2 = 1 2 = px (−1)py (1), so X and Y are not independent. 3 9 3 3 b. 1 1 1 µx = (−1)( ) + (0)( ) + (1)( ) = 0 3 3 3 1 2 2 µy = (0)( ) + (1)( ) = 3 3 3 1 1 1 E(XY ) = (−1)(1)( ) + (0)(0)( ) + (1)(1)( ) = 0 3 3 3 Cov(X, Y ) = E(XY ) − µx µy = 0 − 0 = 0 ⇒ ρ(X, Y ) = 0, X and Y are uncorrelated. 46. In general, Def . Def . 2 Cov(X, X ) = E(X − µx )(X − µx ) = E(X − µx )2 = σx For discrete case specifically, Def . Cov(X, X ) = ￿ x (x − µx )(x − µx )p(x) = ￿ x Def . 2 ( x − µ x ) 2 p( x ) = σ x...
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