Continuous Random Variables

A continuous random variable can take infinitely many values, and its outcomes cannot be counted individually. Instead of listing outcomes, we describe its behavior using a probability density function (pdf) where the area under the curve represents probabilities.

Density Curves

A density curve must satisfy: 1. The total area under the curve equals 1. 2. Every point on the curve has a non-negative height (i.e., the curve does not fall below the \(x\)-axis).

Common Continuous Distributions

Uniform Distribution

A continuous random variable has a uniform distribution if its values are evenly spread over the interval \([a,b]\). Its density function is given by:

\[ \begin{align} f(x)=\frac{1}{b-a},\quad \text{for } a\le x\le b \end{align} \]

The mean and standard deviation are:

\[ \begin{align} \mu &= \frac{a+b}{2},\\[1em] \sigma &= \frac{b-a}{\sqrt{12}} \end{align} \]

Normal Distribution

The normal distribution is symmetric and bell-shaped, described by the equation:

\[ \begin{align} f(x)=\frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \end{align} \]

Here, \(\mu\) is the mean and \(\sigma\) is the standard deviation. A special case with \(\mu=0\) and \(\sigma=1\) is called the standard normal distribution.

Exponential Distribution

The exponential distribution, often referred to as the waiting-time distribution, has the density function:

\[ \begin{align} f(x)=\frac{1}{\theta} \exp\left(-\frac{x}{\theta}\right),\quad \text{for } x\ge0 \end{align} \]

In this case, both the mean and the standard deviation are equal to \(\theta\):

\[ \begin{align} \mu &= \theta,\\[1em] \sigma &= \theta \end{align} \]