Hypoexponential Distribution

Definition

Let $X_1,\dots ,X_n$ be independent exponential random variables with distinct rates ${\lambda }_1,\dots ,{\lambda }_n$ (i.e., ${\lambda }_i\ne {\lambda }_j$ for $i\ne j$).

The Hypoexponential Random Variable is their sum:

$S={\sum }_{i=1}^n{X}_i$

PDF: $f_S(t)={\sum }_{i=1}^n{C}_{i,n}{\lambda }_i{e}^{-{\lambda }_{i}t}$

where: $C_{i,n}={\prod }_{j\ne i}\frac{{\lambda }_j}{{\lambda }_j-{\lambda }_i}$

Tail distribution: $P(S>t)={\sum }_{i=1}^n{C}_{i,n}{e}^{-{\lambda }_{i}t}$

Interpretation

The hypoexponential is the total time to pass through $n$ sequential stages, each with its own exponential rate. Unlike the gamma distribution (which requires identical rates), this handles different rates at each stage.

Asymptotic Behavior

As $t\to \infty$: $r_S(t)\to {\min }_i{\lambda }_i$

The remaining lifetime of a hypoexponentially distributed item that has survived to large age $t$ is approximately exponential with rate equal to the minimum of the component rates.

Remark

The coefficients $C_{i,n}$ are not probabilities — some will be negative. Despite the similar form to the hyperexponential density, these two distributions are very different.

Related

• Exponential Distribution
• Hyperexponential Distribution
• Coxian Distribution