Given the variance of two legs in a population of right triangles, can you find the variance of the hypotenuses?

Question

Let's call the population of the first leg of the right triangles X, the second leg Y, and the hypotenuses H. If I tell you Var(X)=v_x and the Var(Y)=v_y, can you tell me Var(H)? Of course, for all variables x,y,h of X,Y, and H respectively, we have that x^2+y^2=h^2

Answer 1

For a function of two variables $f(x,y)$, from propagation of errors we have $$\sigma_f=\sqrt{\left(\frac{\partial f}{\partial x}\right)^2\sigma_x^2+\left(\frac{\partial f}{\partial y}\right)^2\sigma_y^2}$$ where $\sigma=\sqrt{Var}$ means a standard deviation.

Here, we have $h=h(x,y)=\sqrt{x^2+y^2}$, so $\frac{\partial h}{\partial x}=\frac{1}{2\sqrt{x^2+y^2}}2x=\frac{x}{h}$ and similarly $\frac{\partial h}{\partial y}=\frac{y}{h}$. Using this in the above formula means $$V_h=\frac{x^2}{h^2}V_x+\frac{y^2}{h^2}V_y$$