Continuous Random Variables

Properties of continuous uniform RVs

• Scenario: Events are equally likely to happen anywhere or anytime in an interval of values

• Shorthand: \(X \sim \text{U}[a,b]\)

\[ f_X(x) = \dfrac{1}{b-a}, \text{ for }a \leq x \leq b \]

\[ F_X(x) = \left\{ \begin{array}{ll} 0 & \quad x<a \quad \\ \dfrac{x-a}{b-a} & \quad a \leq x \leq b\quad \\ 1 & \quad x>b \quad \end{array} \right. \]

\[\text{E}(X) = \dfrac{a+b}{2} \text{, } \text{ Var}(X) = \dfrac{(b-a)^2}{12}\]

Properties of exponential RVs

• Scenario: Modeling the time until the next (first) event

• Continuous analog to the geometric distribution!

• Shorthand: \(X \sim \text{Exp}(\lambda)\)

\[ f_X(x) = \lambda e^{-\lambda x}\text{ for } x>0, \lambda>0 \]

\[ F_X(x) = \left\{ \begin{array}{ll} 0 & \quad x<0 \quad \\ 1 - e^{-\lambda x} & \quad x\geq0 \\ \end{array} \right. \]

\[\text{E}(X) = \dfrac{1}{\lambda}\] \[\text{Var}(X) = \dfrac{1}{\lambda^2}\]

Properties of gamma RVs

• Scenario: Modeling the time until the \(r^{th}\) event.
• Continuous analog to the Negative Binomial distribution
• Shorthand: \(X \sim \text{Gamma}(r, \lambda)\)

\[ f_X(x) = \dfrac{\lambda^r}{\Gamma(r)}x^{r-1} e^{-\lambda x}\text{ for } x>0, \lambda>0, \Gamma(r) = (r-1)! \]

\[ F_X(x) = \left\{ \begin{array}{ll} 0 & \quad x<0 \quad \\ 1 - e^{-\lambda x}\displaystyle\sum_{j=0}^{r-1}\dfrac{(\lambda x)^j}{j!} & \quad x\geq0 \\ \end{array} \right. \]

\[\text{E}(X) = \dfrac{r}{\lambda}\text{, }\text{ Var}(X) = \dfrac{r}{\lambda^2}\]

Common to see \(\alpha = r\) and \(\beta = \lambda\)

Properties of Normal RVs

• No scenario description here because the Normal distribution is so universal

  • Central Limit Theorem (next class) makes it applicable to many types of events

• Shorthand: \(X \sim \text{Normal}(\mu, \sigma^2)\)

\[ f_X(x) = \dfrac{1}{\sqrt{2\pi \sigma^2}}e^{-(x-\mu)^2/(2\sigma^2)} \text{, for} -inf < x < inf \]

\[\text{E}(X) = \mu \] \[\text{Var}(X) = \sigma^2\]