Cheat Sheet (Self Learning)

Cheat Sheet (Self Learning)#

Recollect/refer the notations from Module 1

  • Reliability is the probability that a part/system/component will function over time period \(t\)

  • Since the “measurement” is now time, the probability distributions must be defined only in \([0,\infty)\)

Basic Probability#

  • \(Pr(A|B) = \frac{Pr(A \cap B)}{Pr(B)}\)

  • If A and B are independent, \(Pr(A \cap B) = Pr(A)Pr(B)\)

    • If A and B are independent, \(Pr(A \cup B) = Pr(A)+Pr(B)\)

General Definitions#

  • ***Reliability \(R(t) = Pr(T>t) = 1-F(t)\)

  • \(f(t) = \frac{dF(t)}{dt} = - \frac{dR(t)}{dt}\)

  • Mean Time To Failure (MTTF) \(= E(T) = \int_0^{\infty} tf(t) dt\)

  • Hazard or Failure Rate function \(\lambda(t) = \frac{f(t)}{R(t)}\)

  • Conditional Reliability \(R(t|T_0) = \frac{R(T_0 + t)}{R(T_0)} \)

  • A typical bathtub curve (Check Fig 2.3 in the material)

Constant Failure Rate#

Essentially, \(\lambda(t)\) is a constant (\(c\)).

  • If \(\lambda(t) = c \implies R(t) = e^{-c t} \implies f(t) = - \frac{dR(t)}{dt} = ce^{-ct}\)

  • MTTF \( = \frac{1}{c}\)

  • \(R(t|T_0) = R(t)\)

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