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Section 9.4 Other "Bell Shaped" distributions

The Normal distribution discussed above is very important when doing statistical analysis. It however is not the only distribution that is symmetrical about the mean and looks like a bell. In this section, we consider two other options--one which is virtually useless and another which is very useful.

For a useless distribution, consider a continuous random variable on the real numbers defined by

\begin{equation*} f(x) = \frac{1/\pi}{1+x^2}. \end{equation*}

A random variable with this probability function is said to be a Cauchy Distribution.

\begin{equation*} \int_{-\infty}^{\infty} \frac{1}{1+x^2} dx = arctan(\infty) - arctan(-\infty) = \pi/2 - (-\pi/2) = \pi. \end{equation*}
\(\pi\)

Now that we have a probability function, it is important to determine its mean and variance. It should be obvious that when doing so using the Cauchy probability function, problems quickly arise. Indeed,

\begin{equation*} \int_{-\infty}^{\infty} x \frac{1}{1+x^2} dx = (1/2) ( \ln( | \infty |) - \ln( | -\infty |) \end{equation*}

which is problematic. Further, for the variance

\begin{equation*} \int_{-\infty}^{\infty} x^2 \frac{1}{1+x^2} dx \end{equation*}

and note that the integrand does not converge to 0 at the endpoints and therefore the integral is automatically considered divergent. Thus it is reasonable to note that the Cauchy distribution has no variance.

On the other hand, there is another bell-shaped distribution that is useful and its random variable can be created by using a mixture of a normal variable and a Cauchy variable. Indeed, suppose Z is a standard normal variable and Y is \(\Chi^2(r)\) with Y and Z independent. Define a new random variable

\begin{equation*} T = \frac{Z}{\sqrt(Y/r)}. \end{equation*}

Then, T is said to have a (Student) t distribution. The good news is that this distribution is useful and its statistics are presented below without proof.