## The normal Gaussian distribution the standard for error distributions

We now turn to the normal or Gaussian distribution, which most readers will have encountered previously - both in other sources and in our discussion of the physical process of diffusion in Chapter 2. For that reason, I will not belabor matters and repeat much of what you already know, but will quickly move on to what I hope are new matters. However, some introduction is required.

The density function for a random variable X that is normally distributed with mean ^ and variance a2 is f w = T2h exp( (3'70)

Note that I could have taken the square root of the variance, but chose to leave it within the square root. A particularly common and useful version is the normal distribution with mean 0 and variance 1; we denote this by N(0, 1) and write X~ N(0, 1) to indicate that the random variable X is normally distributed with mean 0 and variance 1. In that case, the probability density function becomes f (x) = l/V^Pexp(—x2/2). Indeed, it is easy to see that if a random variable Y is normally distributed withmean ^ and variance a then the transformed variables X = (Y — a will be N(0, 1); we can make a normal random variable Y with specified mean and variance from a X~ N(0, 1) by setting Y = ^ + aX.

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