# Confidence Intervals with Propagation of Uncertainty

Lets say I'm trying to make a measurement of the area, $A$ of an object imaged in a large number of noisy gray-scale image, and I want to include uncertainty quantification to some confidence interval,$X$, $\delta_{CI:\ X}A$.

I'm choosing to measure the area using the formula $A = \mu_L\times \mu_H$ where $L$ is the length and $H$ is the height. The uncertainty in the height and length are $\delta_{CI:\ 0.95}L$ and $\delta_{CI:\ 0.95}H$ assuming they are each normally distributed and independent.

I can calculate the propagation of uncertainty and write it as the equation:

$$\delta_{CI:\ X}A =\sqrt{ \mu_H^2\delta_{CI:\ 0.95}L^2 + \mu_L^2\delta_{CI:\ 0.95}H^2}$$

so now I have the uncertainty of $A$ in terms of $H$, $L$, $\delta_{CI:\ 0.95}L$, and $\delta_{CI:\ 0.95}H$. It is my understanding that $\delta_{CI:\ 0.95}H$ means that I am 95% confident the average lies within $\pm \delta_{CI:\ 0.95}H$ of $\mu_H$ and the same is true for $L$.

So if I am 95% certain in $\mu_L$, and 95% certain in $\mu_H$ am I 95% certain in $A$?

or if I'm 95% confident in each am I instead 90.25% confident (95%^2 = 90.25%) or should I be 99.75% confident (100 - (100-95%)^2 = 99.75%)?

These other options make sense because I've heard the 95% confidence interval described as the expect range of variation in the mean 95% of the time with more sampling.

I'm trying to find out if the confidence level propagates and if it does, how does it change?

• Could you clarify how we are supposed to understand the phrase "some function $F$ ... according to $F=X\times Y$"? Is this just an elliptical way of stating that $F(X,Y)=XY$, or is it intended to mean something else? And since the answer likely depends on your data, your model, and your procedure for computing confidence intervals, please edit your post to include descriptions of those. – whuber Jun 24 '18 at 18:11
• @whuber The $F(X,Y) = XY$ is what I intended, but the functional form shouldn't matter, I'm interested in what is the resultant confidence level of $F$ and its relationship to the confidence levels of $X$ and $Y$ – James Urban Jun 24 '18 at 18:27
• There is no general rule. You should work out the distribution of F from scratch using the knowledge of the formula $f$ and the assumed distributions for X, Y (and also how/if they correlate) – Sextus Empiricus Jun 24 '18 at 18:27
• @MartijnWeterings I can do that but is is there no way to say what the confidence level is for $F$? – James Urban Jun 24 '18 at 18:28
• The functional form matters. Also the distribution of the pairs X,Y matters. – Sextus Empiricus Jun 24 '18 at 18:29