Notes › EENG 3421: Probability and Statistics for Engineers Lecture 18
Families of Continuous Random Variables
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Uniform $(a,b)$
Definition 18.1 (Uniform $(a,b)$ Random Variable).
$X$ is a uniform $(a,b)$ random variable if the PDF of $X$ has the form
$$ f_X(x)=\begin{cases} \dfrac{1}{b-a} & a\le xwhere $b>a$Theorem 18.2 (Uniform CDF Mean Variance).
If $X$ is uniform $(a,b)$
$$ F_X(x)=\begin{cases} 0 & x\le a \\ \dfrac{x-a}{b-a} & aExponential $(\lambda)$
Definition 18.3 (Exponential $(\lambda)$ Random Variable).
$X$ is an exponential $(\lambda)$ random variable if the PDF of $X$ is
$$ f_X(x)=\begin{cases} \lambda e^{-\lambda x} & x\ge 0 \\ 0 & \text{otherwise} \end{cases} $$where $\lambda>0$
Theorem 18.4 (Exponential CDF Mean Variance).
If $X$ is exponential $(\lambda)$
$$ F_X(x)=\begin{cases} 1-e^{-\lambda x} & x\ge 0 \\ 0 & \text{otherwise} \end{cases} $$$$ E[X]=\frac{1}{\lambda} $$$$ \mathrm{Var}[X]=\frac{1}{\lambda^2} $$Erlang $(n,\lambda)$
Definition 18.5 (Erlang $(n,\lambda)$ Random Variable).
$X$ is an Erlang $(n,\lambda)$ random variable if the PDF of $X$ is
$$ f_X(x)=\begin{cases} \dfrac{\lambda^n x^{n-1} e^{-\lambda x}}{(n-1)!} & x\ge 0 \\ 0 & \text{otherwise} \end{cases} $$where $\lambda>0$ and $n\ge 1$ is an integer
Theorem 18.6 (Erlang Mean Variance).
If $X$ is Erlang $(n,\lambda)$
$$ E[X]=\frac{n}{\lambda} $$$$ \mathrm{Var}[X]=\frac{n}{\lambda^2} $$References
- Course lecture slides 18–20
Sources
- Course lecture slides 18–20