Contineous Distributions

Contineous random variables can be used to describe phenomena where the values of a random variable can be any value in an interval: the time to failure, or the length of a broken rod. Seven distributions will be discussed.

1. Uniform distributin.
   • pdf:
     
     $$f(x) = \left\{ \begin{array}{ll} \frac{1}{b-a} & a \le x \le b \\ 0 & {\rm otherwise} \\ \end{array} \right.$$
     
     

   • cdf:
     
     $$F(x) = \left\{ \begin{array}{ll} 0 & x < a \\ \frac{x-a}{b-a} & a \le x \le b \\ 1 & x > b \\ \end{array} \right.$$
     
     

   • mean:
     
     $$E(X) = \frac{a+b}{2}$$
     
     

   • variance:
     
     $$V(X) = \frac{(b-a)^2}{12}$$
     
     

   • the interval where $X$ can assume value can be arbitrarily long, but it cannot be infinite.
   • Example 6.15 and 6.16 on page 202, 203

2. Exponential distribution. Exponential distributed random variable is one of most frequently used distribution in computer simulation. It is widely used in simulations of computer network and others.
   • pdf
     
     $$f(x) = \left\{ \begin{array}{ll} \lambda e^{-\lambda x} & x \ge 0 \\ 0 & {\rm otherwise} \end{array} \right.$$
     
     

   • cdf
     
     $$F(x) = \left\{ \begin{array}{ll} 1 - e^{-\lambda x} & x \ge 0 \\ 0 & x < 0 \\ \end{array} \right.$$
     
     

   • mean
     
     $$E(X) = \frac{1}{\lambda}$$
     
     

   • variance
     
     $$V(X) = \frac{1}{\lambda^2}$$
     
     

   • memoryless property of the exponential distributed random variables: the future values of the exponentially distributed values are not affected by the past values. Compare this to, for example, a uniformly distributed random variable, one can see the difference. For example, when throwing a fair coin, we can consider the probability of head and tail is the same which has the value of 0.5. If, after a result of head, we would expect to see a tail (though it may not happen). In exponentially distributed random variable, we cannot have this type of expectation. In another word, we know nothing about the future value of the random variable given a full history of the past.

     Mathematical proof.
     

     $$P(X > s+t \vert X > s) = \frac{P(X > s+t)}{P(X > s)} = \frac{e^{-\lambda (s+t)}}{e^{-\lambda s}} = e^{-\lambda t} = P(X > t)$$
     
     

   • Example 6.17 and 6.18 on page 204 and 205 where Example 6.18 demonstrates the memoryless property of the exponential distribution.

3. Gamma distribution.
   • pdf
     
     $$f(x) = \left\{ \begin{array}{ll} \frac{\beta \theta}{\Gamma ... ...a x} & x > 0 \\ 0 & {\rm otherwise} \\ \end{array} \right.$$
     
     
     where $\Gamma(\beta) = (\beta - 1)!$ when $\beta$ is an integer.
   • When $\beta == 1$, this is the exponential distribution. In another word, the Gamma distribution is a more general form of exponential distribution.
   • mean
     
     $$E(X) = \frac{1}{\theta}$$
     
     

   • variance
     
     $$V(X) = \frac{1}{\beta \theta^2}$$
     
     

4. Erlang distribution.
   • When the parameter $\beta$ in Gamma distribution is an integer, the distribution is refered to as Erlang distribution.
   • When $\beta = k$, a positive integer, the cdf of Erlang distribution is (using integration by parts)
     
     $$F(x) = \left\{ \begin{array}{ll} 1 - \sum_{i=0}^{k-1} \frac{... ...eta x)^i}{i!} & x > 0 \\ 0 & x\le 0 \\ \end{array} \right.$$
     
     
     which is the sum of Poisson terms with mean $\alpha = k \theta x$
   • mean
     
     $$E(X) = \frac{1}{\theta}$$
     
     

   • variance
     
     $$V(X) = \frac{1}{k \theta^2}$$
     
     

   • Example 6.19, 6.20 on page 208, 209.

5. Normal distribution.
   • pdf
     
     $$f(x) = \frac{1}{\sigma \sqrt{2 \pi}} exp\left[ - \frac{1}{2} ... ...eft(\frac{x-\mu}{\sigma}\right)^2\right], -\infty < x < \infty$$
     
     

   • cdf
     
     $$F(x) = P(X \le x) = \int_{-\infty}^x \frac{1}{\sqrt{2\pi}} ex... ...\frac{1}{2} \left( \frac{t - \mu}{\sigma} \right)^2 \right] dt$$
     
     

     This value is very difficult to calculate. Often a table is made for $X \sim N(0,1)$. Because an $X \sim N(\mu, \sigma^2)$ can be transformed into $X \sim N(0,1)$ by let
     

     $$Z = \frac{X-\mu}{\sigma}$$
     
     

   • To calculate $P(X \le x)$ for $X \sim N(\mu, \sigma^2)$, we use $\Phi (\frac{x - \mu}{\sigma})$

     Example: to calculate F(56) for N(50,9), we have
     

     $$F(56) = \Phi (\frac{56 - 50}{3}) = \Phi(2) = 0.9772$$
     
     

   • mean $\sigma$
   • variance $\sigma^2$
   • Notation: $X \sim N(\mu, \sigma^2)$
   • The curve shape of the normal pdf is like a "bell".
   • properties:
     • $lim_{x \rightarrow -\infty}f(x) = 0$ and $lim_{x \rightarrow \infty}f(x) = 0$
     • $f(\mu - x) = f(\mu + x)$ the pdf is symmetric about $\mu$ because of this, $\Phi(-x) = 1 - \Phi(x)$.
     • the maximum value of the pdf occurs at $x=\mu$ (thus, the mean and the mode are equal.

   • Example 6.21 and 6.22 on page 211, 6.23 and 6.24 on page 213.

6. Weibull distribution. The random variable $X$ has a Weibull distribution if its pdf has the form
   
   $$f(x) = \left\{ \begin{array}{ll} \frac{\beta}{\alpha} \left( ... ...ht] & x \ge \gamma \\ 0 & \rm {otherwise} \end{array} \right.$$
   
   
   • Weibull distribution has the following three parameters:
     1. $\gamma$ which has the range of $-\infty < \gamma < \infty$ which is the location parameter
     2. $\alpha$ which is greater than zero which is the scale parameter
     3. $\beta$ which is a positive value determines the shape