## The lognormal distribution and nonnegative measurements

We are still not done with the normal distribution and its variants, because very often we take measurements that can only have positive values. Thus our very first, and simplest model, in Eq. (3.75), Y = m + X- where m is fixed but unknown and X are N(0, 1), will fail when the observed values are required by biological or physical law to be positive (e.g. masses, lengths, or gene frequencies).

The way to avoid breaking natural law is to use the log-normal distribution, which we construct as follows. First, note that if

X~ N(0, 1), then a X will be normally distributed with mean 0 and variance a2. We then define a new random variable Y by

so that log( Y) is normally distributed with mean log(A) and variance a2. Now, althoughXtakes values from —i to i, the exponential will only take values from 0 upwards. Thus, although E{X} = 0, we conclude that E{Y} > A. As it happens, we can compute all of the moments for Y in one calculation, which is pretty snazzy. That is, let us consider

so that we have to compute i

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