ncy components. For instance, we have this nifty equation (relating a normal distribution to its Fourier transform):
Note that the general equation for a (1 dimensional) Gaussian distribution (which tells us the likelihood of each value x) is
where is the mean of the distribution, and is its standard deviation. Hence, the Fourier transform relation above deals with a normal distribution of mean 0 and standard deviation .
Another useful property to take note of relates to solving maximum likelihood problems (where we are looking for the parameters that make some data set as likely as possible). We generally end up solving these problems by trying to maximize something related to the log of the probability distribution under consideration. If we use a normal distribution, this takes the unusually simple form
which is often nice enough to allow for solutions that can be calculated exactly by hand. In particular, the fact that this function is quadratic in x makes it especially convenient, which is one reason that the Gaussian is commonly chosen in statistical modeling. In fact, the incredibly popular ordinary least squares regression technique can be thought of as finding the most likely line (or plane, or hyperplane) to fit a dataset, under the assumption that the data was generated by a linear equation with additive gaussian noise.