How to change variables elegantly?

How can I better quantify the change to outcome likelihood when model variables are moved away from their known mean value?

• Assume I have two business model variables - feedstock cost (for making widgets) and widget sale price.  Each has a mean of x and a STD of y.  A business is modeled with feedstock cost assumed to be x, and widget sale price assumed to be x.  This model does not show a positive outcome. To counter this, non-quantifiable goals are set for "cost control" and "sales tactics" with the result being that the assumed feedstock price is changed to x-y, and the assumed widget sale price is changed to x+y.  Now each variable is one standard deviation from the mean in the direction that leads to a more favorable outcome (lower feedstock price and higher widget sale price) Both variables are still reasonable assumptions, but risk has been introduced by deviating from the known mean value.  How can I quantify this increased risk if I know the mean and STD for each critical variable?

• Answer:

  Okay, so here's the problem as I understand it: We have a business with revenue R and cost C, where R and C are random variables with the same mean x and standard deviation y; consequently, the expected profit of this operation is zero. This doesn't appeal to the management, who'd like to fudge the numbers so that the expected revenue is x + y and the expected costs are x - y instead,  leading to an expected profit of 2y with some greater uncertainty. Unfortunately -- or fortunately, from the perspective of outside investors -- you simply can't do this.  The expected profit is a function of the distribution of profits, which is a function of the revenue and cost distributions; we can't somehow change the description of the distribution in such a way that the expectation is different but the uncertainty increases. There is one way that an estimate like this could conceivably be salvaged, but be warned that it probably doesn't lie within the body of generally-accepted accounting practices.  I present the following, in the words of Leo Bloom, purely for academic interest: Before, we said that the revenue and cost each had expectation x and standard deviation y.  Let's take that back; we didn't really mean it.  What we actually meant was that there was a 68% chance that the revenue would fall between x - y and x + y (as there would be for a normal distribution with expectation x and standard deviation y), but that the expectation of the distribution was in fact x + y.  We'd then make the analogous assertion about the costs as well.  For the sake of making the calculations easier, we could assume that the revenue and costs follow skew-normal distributions, which we could then calibrate so that these results hold. Of course, to fit with the previous results, you will be installing a substantial amount of skew to compensate -- instead of a model with roughly symmetric wins and losses, you'll be building a model with a large chance of small wins and a huge chance of small losses, which is the proverbial "picking up nickles in front of a steamroller" scenario.  It's not hard to see why -- your business plan is to hope for two simultaneous one-standard-deviation wins, which is about a 40:1 long shot assuming normal errors. When I get a chance later, I'll try actually calibrating the skew-normal models so as to provide some pictures of what happens here.