Fubini’s theorem
Topic of Reliability, part of Probability
Fubini’s theorem
| Type | Topic |
| Taught in | Reliability |
| Part | Probability |
For a non-negative function, or one whose absolute value is integrable, a double integral or a double sum may be computed in either order. Applied to \mathbb{E}T=\mathbb{E}\int_0^T1\,dt it gives \mathbb{E}T=\int_0^\infty\mathbb{P}(T>t)\,dt for a non-negative T.
In Reliability
From Reliability
For a non-negative random variable the expectation is the integral of the survival function, \mathbb{E} T=\int_0^{\infty} R(t)\,\,\mathrm{d} t, when the integral exists. The argument is integration by parts, or Fubini’s theorem applied to \mathbb{E} T=\mathbb{E}\int_0^{T} 1\,\,\mathrm{d} t=\int_0^{\infty}\mathbb{P}(T>t)\,\,\mathrm{d} t. The mean time to failure is this expectation.
Used in
- Taught in Reliability
- All 7 questions on Reliability
See also
Structure function · Reliability function · Survival function · Hazard rate · Weibull distribution