statisticalrethinkin..

86 4. LINEAR MODELSexperiment with even larger numbers of steps to verify for yourself that the distribution ofpositions is stabilizing on the Gaussian. You can square the step sizes and transform themin a number of arbitrary ways, without changing the result: normality emerges. Where doesit come from?Any process that adds together random values from the same distribution converges toa normal. But it’s not easy to grasp why addition should result in a bell curve of sums. 53Here’s a conceptual way to think of the process. Whatever the average value of the sourcedistribution, each sample from it can be thought of as a fluctuation from that average value.When we begin to add these fluctuations together, they also begin to cancel one another out.A large positive fluctuation will cancel a large negative one. e more terms in the sum, themore chances for each fluctuation to be cancelled by another, or by a series of smaller onesin the opposite direction. So eventually the most likely sum, in the sense that there are themost ways to realize it, will be a sum in which every fluctuation is cancelled by another, asum of zero (relative to the mean). 54It doesn’t matter what shape the underlying distribution possesses. It could be uniform,like in our example above, or it could be (nearly) anything else. 55 Depending upon the underlyingdistribution, the convergence might be slow, but it will be inevitable. Oen, as inthis example, convergence is rapid.4.1.2. Normal by multiplication. Here’s another way to get a normal distribution. Supposethe growth rate of an organism is influenced by a dozen loci, each with several alleles thatcode for more growth. Suppose also that all of these loci interact with one another, such thateach increase growth by a percentage. is means that their effects multiply, rather than add.For example, we can sample a random growth rate for this example with this line of code:R code4.2prod( 1 + runif(12,0,0.1) )is code just samples 12 random numbers between 1.0 and 1.1, each representing a proportionalincrease in growth. us 1.0 means no additional growth and 1.1 means a 10%increase. e product of all twelve is computed and returned as output. Now what distributiondo you think these random products will take? Let’s generate 10-thousand of them andsee:R code4.3growth