Hyperexponential Distribution

Definition

Let $X_1,\dots ,X_n$ be independent exponential random variables with respective rates ${\lambda }_1,\dots ,{\lambda }_n$. Let $T$ be a discrete random variable independent of these, with $P(T=j)=P_j$ and ${\sum }_{j=1}^n{P}_j=1$.

The Hyperexponential Random Variable is $X=X_T$, i.e., a mixture of exponentials:

$1-F(t)={\sum }_{j=1}^n{P}_j{e}^{-{\lambda }_{j}t}$

Density: $f(t)={\sum }_{j=1}^n{\lambda }_j{P}_j{e}^{-{\lambda }_{j}t}$

Interpretation

Imagine a bin with different types of batteries, each type having its own exponential lifetime. You pick one at random — the lifetime of your chosen battery is hyperexponential. It’s a “mixture” model.

Failure Rate

$r(t)={\sum }_{j=1}^n{\lambda }_{j}P(T=j\mid X>t)$

As $t\to \infty$: $r(t)\to {\min }_j{\lambda }_j$

The failure rate converges to the smallest rate (longest-lived type), because as the item ages, it becomes increasingly likely to be the most durable type.

Related

• Exponential Distribution
• Hypoexponential Distribution
• Failure Rate Function