# What is the variance of Z? if $Z = \frac{X}{Y}$ and X and Y are given.

We measure voltage across a resistor and find that the measured samples are distributed normally with mean $$E[X]$$ and variance $$\sigma_X$$, we measure current across a resistor and find that the measured samples are distributed normally with mean $$E[Y]$$ and variance $$\sigma_Y$$. It is given that both measurements are independent of each other. Random variables X and Y are independent.

What is the mean and variance of Z which represents the resistance? Since $$V = IR$$,

$$Z = \frac{X}{Y}$$

This question is inspired from the GATE 2019 Instrumentation paper which had the following question,

So I would like to specifically find the $$Var(Z)$$ if $$E[X] = 1$$, $$E[Y] = 10^{-3}$$, $$Var(X) = (0.12)^2$$ and $$Var(Y) = (0.05 \times 10^{-3})^2$$

## My attempts

From my googling, I find that

There is no formula expressing $$E[\frac{1}{Y}]$$ or $$var(\frac{1}{Y})$$ in terms of $$E[Y]$$ and $$var(Y)$$

So my initial attempts as shown below failed.

$$var(XY)=E(X^2Y^2)−E(XY)^2=var(X)var(Y)+var(X)E(Y)^2+var(Y)E(X)^2$$

The pdf is not given to us, so I can't manually integrate and find the expectation.

Since this question appeared in a prestigious national exam, I am confident a solution exists. Further this question was only worth 1 mark so I am expecting an elegant solution to exist.

Can we think of standard deviation as percent error? and simply add it?

• What level is the exam? They might just want you to apply basic error propagation... – Thomas Feb 8 at 13:47
• It is meant for recently graduated students. GATE. I think you are right, it simply needs adding of the error I think. I can't believe I did not think of that. :( – Aditya Feb 8 at 13:50
• Would and approximation be also good enough? – Math-fun Feb 8 at 13:57
• For the record, no normal random variable $Y$ with nonzero variance (that is, not Dirac) is integrable. Thus, $E(1/Y)$, $E(1/Z)$, $\mathrm{var}(Y)$ and $\mathrm{var}(Z)$ are all undefined. GATE 2019 asking this is a display of incompetence. – Did Feb 8 at 14:19
• Mmm... For example here en.m.wikipedia.org/wiki/Propagation_of_uncertainty look at the two variable example . It is essentially what J.G. proposes as the delta method, but such formulas are usually presented in lab courses via intuitive approaches – Thomas Feb 9 at 15:17

What you're expected to do here is use the delta method, affecting the approximation $$\operatorname{Var}f(X_i)\approx\sum_i\left(\frac{\partial f}{\partial X_i}\right)^2\operatorname{Var}X_i$$. In this case, $$\sigma_Z^2\approx\frac{\sigma_X^2}{Y^2}+\frac{X^2\sigma_Y^2}{Y^4}$$.
• For the record you mean $E[X]^2$ when you say $X^2$ right? So that comes to $\sigma_Z = \sqrt{\frac{0.12^2}{10^-6} + \frac{1^2(0.05 \times 10^{-3})^2}{10^{-12}}} = 130$ – Aditya Feb 8 at 15:00