求解一道概率题设随机变量X1,X2,…,Xn相互独立,D(Xi)=δi^2,δi不等于0,i=1,2…,n.又∑(i从1到n)ai=1,求ai(i=1,2…,n),使∑(i从1到n)aiXi的方差最小.答案提示用构造拉格朗日函数L=∑(i从1到n)(a
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![求解一道概率题设随机变量X1,X2,…,Xn相互独立,D(Xi)=δi^2,δi不等于0,i=1,2…,n.又∑(i从1到n)ai=1,求ai(i=1,2…,n),使∑(i从1到n)aiXi的方差最小.答案提示用构造拉格朗日函数L=∑(i从1到n)(a](/uploads/image/z/5910347-11-7.jpg?t=%E6%B1%82%E8%A7%A3%E4%B8%80%E9%81%93%E6%A6%82%E7%8E%87%E9%A2%98%E8%AE%BE%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8FX1%2CX2%2C%E2%80%A6%2CXn%E7%9B%B8%E4%BA%92%E7%8B%AC%E7%AB%8B%2CD%EF%BC%88Xi%EF%BC%89%3D%CE%B4i%5E2%2C%CE%B4i%E4%B8%8D%E7%AD%89%E4%BA%8E0%2Ci%3D1%2C2%E2%80%A6%2Cn.%E5%8F%88%E2%88%91%EF%BC%88i%E4%BB%8E1%E5%88%B0n%EF%BC%89ai%3D1%2C%E6%B1%82ai%28i%3D1%2C2%E2%80%A6%2Cn%29%2C%E4%BD%BF%E2%88%91%EF%BC%88i%E4%BB%8E1%E5%88%B0n%EF%BC%89aiXi%E7%9A%84%E6%96%B9%E5%B7%AE%E6%9C%80%E5%B0%8F.%E7%AD%94%E6%A1%88%E6%8F%90%E7%A4%BA%E7%94%A8%E6%9E%84%E9%80%A0%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E5%87%BD%E6%95%B0L%3D%E2%88%91%EF%BC%88i%E4%BB%8E1%E5%88%B0n%EF%BC%89%EF%BC%88a)
求解一道概率题设随机变量X1,X2,…,Xn相互独立,D(Xi)=δi^2,δi不等于0,i=1,2…,n.又∑(i从1到n)ai=1,求ai(i=1,2…,n),使∑(i从1到n)aiXi的方差最小.答案提示用构造拉格朗日函数L=∑(i从1到n)(a
求解一道概率题
设随机变量X1,X2,…,Xn相互独立,D(Xi)=δi^2,δi不等于0,i=1,2…,n.又∑(i从1到n)ai=1,求ai(i=1,2…,n),使∑(i从1到n)aiXi的方差最小.
答案提示用构造拉格朗日函数L=∑(i从1到n)(aiδi)^2+λ(∑(i从1到n)ai-1)=0;
∑(i从1到n)ai=1.然而不会解离散型变量的拉格朗日的这个方程..
求解一道概率题设随机变量X1,X2,…,Xn相互独立,D(Xi)=δi^2,δi不等于0,i=1,2…,n.又∑(i从1到n)ai=1,求ai(i=1,2…,n),使∑(i从1到n)aiXi的方差最小.答案提示用构造拉格朗日函数L=∑(i从1到n)(a
因为X1,X2,…,Xn相互独立,所以
D(∑(i从1到n)aiXi) = ∑(i从1到n)D(aiXi) = ∑(i从1到n)ai^2 D(Xi) = ∑(i从1到n)ai^2 δi^2
设 L(a1,...,an,λ) = ∑(i从1到n)(aiδi)^2+λ(∑(i从1到n)ai-1),
当给定 a1,...,a(i-1),a(i+1),...,an,λ时,L是ai的二次函数,且开口向上.
于是在最小值处,有:
下面用 dL/dai 表示偏导数.
dL/dai = 2ai δi^2 + λ = 0 ,i = 1,...,n
==> -λ/2 = a1 δ1^2 = a1/(1/ δ1^2) = .= an/(1/ δn^2)
= (a1 + .+an)/((1/ δ1^2) + ...+(1/ δn^2))
= 1/ ((1/ δ1^2) + ...+(1/ δn^2))
==>
ai = -λ / (2δi^2) = 1/δi^2 * (-λ/2)= 1/δi^2 / ((1/ δ1^2) + ...+(1/ δn^2)) ,i = 1,2,...,n
当 ai ,i=1,...,n,为上值时,方差最小.