Taylor approximation¶
first-order¶
\[f(x)\approx f(a)+f'(a)(x-a)\]
so the approximation of \(f(x)=\log(1+x)\) around \(x=0\) is
\[\log(1+x)\approx x\]
second-order¶
\[f(x)\approx f(a)+f'(a)(x-a)+\dfrac{1}{2}f''(x)(x-a)^2\]
example¶
given \(g(X)\) & \(\mu=E(X)\)
first-order
\[g(X)\approx g(\mu)+g'(\mu)(X-\mu)\]
\[Var(g(X))=[g'(\mu)]^2Var(X)\]
second-order
\[g(X)\approx g(\mu)+g'(\mu)(X-\mu)+\dfrac{1}{2}g''(X)(X-\mu)^2\]
\[E[g(X)]\approx g(\mu)+\dfrac{1}{2}g''(\mu)Var(X)=g(\mu)+\dfrac{1}{2}\dfrac{g''(\mu)}{[g'(\mu)]^2}Var(g(X))\]
when \(g(X)=\log(X)\)
\[E[\log(X)]\approx \log(\mu)-\dfrac{1}{2}Var(\log(X))\]
\[\log E(X)=E(\log X)+\dfrac{1}{2}Var(\log X)\]
\[E(X)=e^{E(\log X)+\frac{1}{2}Var(\log X)}\]