# Using probability distributions and variances to modify a formula

We have a lamp and it is in 2m distance. Its light intensity is 5. We want to know what amount of light is reaching to us. We can simply use a formula like: $\mathrm{light} = \frac{\mathrm{intensity}}{\mathrm{distance}}$.

Now suppose that the distance is a random variable (i.e. it is in $2m \pm 1m$ distance, and the intensity is $5 \pm 4.7$).

How we can change the formula in order to be able to use it in case we have random variables too?

P.S. Actually the question is how we can make use of probability distributions, variance, etc. to change the formula?

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I would guess that the denominator should be squared unless something coherent is going on... –  copper.hat Jul 6 '12 at 21:03
can't we work and make use of probability distributions here? –  AKH Jul 6 '12 at 21:07
And there is no need to use three exclamation points! There was a time when typing an exclamation point required 3 keystrokes: you typed a vertical bar, a backspace, and then a period. That made people be somewhat parsimonious with their use. Maybe we should ask that anybody under a certain age be forced to use five or six keystrokes before typing one? –  Arturo Magidin Jul 6 '12 at 21:08
@ArturoMagidin: believe me, people of all ages are prone to this particular phenomenon!! –  Ben Millwood Jul 6 '12 at 21:10
@Ben: And yet, it seems to infect the young more; multiple exclamation points, multiple question marks... Sigh. –  Arturo Magidin Jul 6 '12 at 21:13

## 1 Answer

Your question seems to be when a variable is a ratio of two measurements each having uncertainty associated with it shouldn't we deal with the ratio like any other random variable? The answer is yes and this is often done. The delta method is one way to approximate the variance of a function of random variables when the variance(s) of the variable(s) that are inputs to the function are known.

In the case of ratios there are special issues. For example if X and Y are independent standard normal random variables X/Y is distributed as a Cauchy random variable which has an infinite variance and also an infinite mean. When the denominator has positive probability density around 0 the heavy tail can mean that the variance is infinite.

When you know the density functions for X and Y and they are independent you can write down the joint density of X and Y apply a change of variables mapping (X,Y) into (X/Y, Y) and integrating out Y.

Also by Jensen's inequality E(X/Y)>=E(X)/E(Y). The inequality is strict unless X/Y or Y have degenerate distributions. So there is a lot that can be done to deal with ratios of random variables.

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