I am given that
$\text{var}(h) = \sum_{i=1}^k (\frac{\partial h}{\partial X_i})^2 \ \sigma_i^2$
and with this I need to prove that:
- $\sigma(X+Y) = \sqrt{\sigma_x^2 + \sigma_y^2}$
- $\lambda$ = $\frac{\sigma}{|\overline{x}|}$
- $\lambda(XY)$ = $\sqrt{\lambda_x^2 + \lambda_y^2}$
- $\lambda_x$$^{\alpha}$ = $|\alpha| \ \lambda_x$
where
- $\sigma$ is the standard error
- $\lambda$ is the fractional standard error
- X, Y are independent variables
- h is a function, $h(X_1, X_2, ... , X_i)$ of random variables
- $\sigma_i^2 = \text{var}(X_i)$
I do not know where to begin, I'm assuming that:
$\text{var}(X+Y) = \sum_{i=1}^k (\frac{\partial (X+Y)}{\partial X_i})^2 \ \sigma_x^2$ + $\sum_{i=1}^k (\frac{\partial (X+Y)}{\partial X_i})^2 \ \sigma_y^2$
but I do not know where to go from here. Some help would be greatly appreciated.
You started well, but I think you don't understand what that formula means. In the first formula, $h$ is a function of some variables $X_i$. In your formula you have renamed them to $X=X_1$ and $Y=X_2$. The sum is not from $1$ to $2$, but instead the sum is over $X$ and $Y$. This means that you should write $$\begin{align}\text{var}(X+Y) &= \left(\frac{\partial (X+Y)}{\partial X}\right)^2 \ \sigma_x^2 + \left(\frac{\partial (X+Y)}{\partial Y}\right)^2 \ \sigma_y^2\\&=1^2\sigma_x^2+1^2\sigma_y^2\end{align}$$