### Abstract

We derive probabilistic generalizations of the fundamental theorem of calculus and Taylor's theorem, obtained by making the argument interval random. The remainder terms are expressed in terms of iterates of the familiar stationary-excess or equilibrium residual-lifetime distribution from the theory of stochastic point processes. The probabilistic generalization of Taylor's theorem can be applied to approximate the mean number of busy servers at any time in an M_{t}/G/∞ queueing system.

Original language | English (US) |
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Pages (from-to) | 51-54 |

Number of pages | 4 |

Journal | Statistics and Probability Letters |

Volume | 16 |

Issue number | 1 |

DOIs | |

State | Published - Jan 4 1993 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Taylor's theorem
- fundamental theorem of calculus
- infinite-server queues
- nonstationary queues
- residual lifetime
- stationary-excess distribution
- stochastic point processes

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## Cite this

Massey, W. A., & Whitt, W. (1993). A probabilistic generalization of Taylor's theorem.

*Statistics and Probability Letters*,*16*(1), 51-54. https://doi.org/10.1016/0167-7152(93)90122-Y