# Summing normal instead of beta distributions, consequences for the density function of the sum?

Background: I've modeled a project effort prediction as a Google Spreadsheet template. Details of the Model: http://sites.google.com/site/effortprediction/methodology .Google Spreadsheet does not implement beta distribution functions. In PERT a beta distribution is used to estimate the effort of a task in Ideal Person Days (i.e.: one worker 8h a day without distraction).

Problem: What consequences has use of normal distributions instead of beta distributions when the distribution are summed up to a single distribution.

Details A project consists of a list of tasks. Each task is beta distributed and independed, the sum of the tasks therefore should obey to the central limit theorem. The project therefore is normal distributed. How does Expected Value and SD change for the project when I use normal instead of beta distributions for each task? I assume that the tasks distribution is skewed more often to the right than to the left.

Questions

• Are the tails fater?
• Is the Expected Value higher or lower with normal distributions?
• Is the dispersion higher or lower?
• If you match the mean and variance of each task with the normal distribution, then the expected value and standard deviation of the sum is unchanged. You will loose all information about skewness though - and tails are likely to be thinner with normal Apr 16, 2011 at 11:00
• mean sounds correct, but not sure with beta, i forgot to say how i calculate sd. I start form 90% confidence interval and mean. now i calculate the sd from a norm dist via z-score for 90%. here an error should already be introduced. but what can I say about the error of sd in comparision with beta-dist? Apr 16, 2011 at 11:25
• so you don't have a "standard deviation" specified, you have specified the quantiles - 5% and 95% I take it. But in this case - if you use a beta distribution to fit the quantiles, and then use a normal to approximate that beta distribution, my comment above applies. Apr 16, 2011 at 12:06
• An exponential distribution may be a better idea for modeling instead of the beta distribution. one because it is highly right skewed, two because it is always positive, three because it has "fat tails", and four because the sum of exponentially distributed quantities has an Erlang or gamma or scaled chi-square distribution - which is no more difficult to handle than a normal, plus it is precise. And exponential arises as a limiting form of a beta distribution. Also exponential has maximum entropy (uncertainty) for a given mean - so you only need to ask for expected time to complete Apr 16, 2011 at 23:06
• @roland - the "memoryless" property is a bit confusing because it is conditional, not absolute. More work/effort would result in a higher rate parameter, and thus a smaller amount of time to finish a project. the memorylessness would be assessed by looking at the distribution of the "late tasks". Now why would a task be "late"? One way I can think of is that the amount of work required to finish it was under-specified - adding in these additional tasks - seems like we are back to the start - so memorylessness may be a good property. Apr 17, 2011 at 14:29