Examples of continuous distributions

A random variable has a cumulative distribution $F_X(x)$ if $P(X \le x) = F_X(x)$. Examples of cumulative distributions and their relative density functions are given below.

1. Exponential density function:
   $$f_X(x) = \left\{ \begin{array}{ll} 0 & \mbox{ if } x < 0, \\ \lambda e^{-\lambda x} & \mbox{ if } x \ge 0. \end{array}\right.$$
   The random variable $X$ has the cumulative distribution (shown in figure 1.3):
   $$F_X(x) = \int_{-\infty}^x f_X(t) dt = (x\ge 0) = \int_0^x \lambd... .... -\frac{\lambda e^{-\lambda t}}{\lambda}\right\vert _0^x = 1 - e^{-\lambda x}$$

   Figure 1.3: Exponential distribution.

   

2. standard Gaussian (Normal) distribution ($N(0,1)$):
   $$f_X(x) = {1 \over \sqrt{2 \pi} } e^{-x^2 /2} %%\label{eq:gaussf}$$
   and
   $$F_X(x) = {1 \over \sqrt{2 \pi} } \int_{-\infty}^{x} e^{-y^2 /2} dy, %%\label{eq:gaussCDF}$$
   called the standard Gaussian distribution.

   If $t=(x-\mu)/\sigma$ then we have the Gaussian distribution with mean $\mu$ and variance $\sigma^2$ ( $N(\mu,\sigma^2)$):

   $$f_X(x) = {1 \over \sqrt{2 \pi} \sigma } e^{-\frac{1}{2} (\frac{x-\mu}{\sigma})^2}$$
   and
   $$F_X(x) = {1 \over \sqrt{2 \pi} \sigma } \int_{-\infty}^{x} e^{-\frac{1}{2} (\frac{y-\mu}{\sigma})^2} dy$$

   Figure 1.4: Gaussian cumulative distribution

   

3. Lognormal distribution:

   let $y=\ln x$ and $y\in N(\mu,\sigma^2)$ is a lognormal random variable. Then, since

   $$\frac{dy}{dx} = 1 \qquad\qquad f_X(x) = f_Y(y) \frac{d y}{dx}$$
   $$f_Y(y) = {1 \over \sqrt{2 \pi} \sigma_y } e^{-\frac{1}{2} (\frac{y-\mu_y}{\sigma_y})^2}$$
   $$f_X(x) = {1 \over \sqrt{2 \pi} \sigma_y x } e^{-\frac{1}{2} (\frac{\ln x-\mu_y}{\sigma_y})^2}$$

   Figure 1.5: Lognormal cumulative distribution