Coxian Distribution

Definition

Let

• $X_1,\dots ,X_m$ be independent exponential random variables with distinct rates ${\lambda }_1,\dots ,{\lambda }_m$.
• $N$ be a discrete random variable independent of these with $P(N=n)=P_n$.

The Coxian Random Variable is:

$Y={\sum }_{j=1}^N{X}_j$

Interpretation

An item goes through up to $m$ stages of treatment. After completing stage $n$, it quits the program with probability $r(n)=P(N=n\mid N\ge n)$. The total time spent in the program is Coxian.

Connection to Discrete Failure Rate

If we interpret $N$ as a lifetime measured in discrete time periods, then:

$r(n)=P(N=n\mid N\ge n)$

is the discrete time failure rate — the probability the item “dies” in period $n$ given it survived to that point. This is the discrete analog of the failure rate function $r(t)$.

Density

Conditioning on $N$: $f_Y(t)={\sum }_{n=1}^m{P}_n{f}_{X_1+\cdots +X_n}(t)$

where $f_{X_1+\cdots +X_n}$ is the hypoexponential density.

Related

• Exponential Distribution
• Hypoexponential Distribution
• Failure Rate Function