设随机变量X和Y独立同服从参数为λ的泊松分布，令U=2X+Y,V=2X−Y，求U和V的协方差Cov(U,V)．

步骤 1：确定随机变量X和Y的期望和方差
由于X和Y独立同服从参数为λ的泊松分布，所以E(X) = E(Y) = λ，Var(X) = Var(Y) = λ。

步骤 2：计算U和V的期望
U = 2X + Y，V = 2X - Y
E(U) = E(2X + Y) = 2E(X) + E(Y) = 2λ + λ = 3λ
E(V) = E(2X - Y) = 2E(X) - E(Y) = 2λ - λ = λ

步骤 3：计算U和V的协方差
Cov(U,V) = E[(U - E(U))(V - E(V))]
= E[(2X + Y - 3λ)(2X - Y - λ)]
= E[(2X - 2λ + Y - λ)(2X - 2λ - Y + λ)]
= E[(2X - 2λ)(2X - 2λ) + (2X - 2λ)(-Y + λ) + (Y - λ)(2X - 2λ) + (Y - λ)(-Y + λ)]
= E[4X^2 - 8Xλ + 4λ^2 - 2XY + 2Xλ + 2XY - 2Xλ - Y^2 + Yλ + Yλ - λ^2]
= E[4X^2 - 8Xλ + 4λ^2 - Y^2 + 2Yλ - λ^2]
= 4E[X^2] - 8λE[X] + 4λ^2 - E[Y^2] + 2λE[Y] - λ^2
= 4(Var(X) + E[X]^2) - 8λ^2 + 4λ^2 - (Var(Y) + E[Y]^2) + 2λ^2 - λ^2
= 4(λ + λ^2) - 8λ^2 + 4λ^2 - (λ + λ^2) + 2λ^2 - λ^2
= 4λ + 4λ^2 - 8λ^2 + 4λ^2 - λ - λ^2 + 2λ^2 - λ^2
= 3λ