𝑃𝐷𝐹(𝑥) = 𝑃(𝑋=𝑥)

expresses the probability that the system fails AT time 𝑡

• a Probability Distribution Function of a DISCRETE random variable is a Probability MASS Function
• a Probability Mass Function 𝑃𝐷𝐹(𝑥) has the following properties:
  • 0 ≤ 𝑃𝐷𝐹(𝑥) ≤ 1 for all 𝑥∊𝑋
  • 𝛴_𝑥∊𝑋𝑃𝐷𝐹(𝑥) = 1

• a Probability Distribution Function of a CONTINUOUS random variable is a Probability DENSITY Function
• a Probability Density Function 𝑃𝐷𝐹(𝑥) has the following properties:
  • 𝑃𝐷𝐹(𝑥) ≥ 0, for 𝑠𝑡𝑎𝑟𝑡 < 𝑥 < 𝑒𝑛𝑑
  • _{𝑠𝑡𝑎𝑟𝑡}∫^𝑒𝑛𝑑𝑃𝐷𝐹(𝑥)𝑑𝑥 = 1

𝐶𝐷𝐹(𝑥) = 𝑃(𝑋≤𝑥)

expresses the probability that the system fails before time 𝑡

𝐶𝐷𝐹(𝑎) = 𝛴_{𝑠𝑡𝑎𝑟𝑡≤𝑥≤𝑎}𝑃𝐷𝐹(𝑥)

𝐶𝐷𝐹(𝑎) = _{𝑠𝑡𝑎𝑟𝑡}∫^𝑎𝑃𝐷𝐹(𝑥)𝑑𝑥

𝑅𝐶𝐷𝐹(𝑥) = 𝑃(𝑋≥𝑥)

expresses the probability that the system is still operational at time 𝑡

𝑅𝐶𝐷𝐹(𝑎) = 𝛴_{𝑎≤𝑥≤𝑒𝑛𝑑}𝑃𝐷𝐹(𝑥)

𝑅𝐶𝐷𝐹(𝑎) = _𝑎∫^𝑒𝑛𝑑𝑃𝐷𝐹(𝑥)𝑑𝑥

𝐻𝑃𝐹(𝑥) = 𝑃𝐷𝐹(𝑥) / 𝑅𝐶𝐷𝐹(𝑥)

expresses risk that the system fails AT time 𝑡
i.e. given that the system is still operational at time 𝑡, what is the probability it fails at time 𝑡

∆𝑡 Dependent Hazard Function

𝐻𝑃𝐹(𝑡, ∆𝑡) = 𝑃(𝑡 ≤ 𝑇 ≤ 𝑡 + ∆𝑡 | 𝑇 ≥ 𝑡)

∆𝑡 Independent Hazard Function

𝐻𝑃𝐹(t) = 𝑙𝑖𝑚_∆𝑡→0 [ 𝑃(𝑡 ≤ 𝑇 ≤ 𝑡 + ∆𝑡 | 𝑇 ≥ 𝑡) / ∆𝑡 ]

𝐻𝑃𝐹(𝑡) = 𝑃𝐷𝐹(𝑡) / ∫_𝑡^∞𝑃𝐷𝐹(𝑡)𝑑𝑡

𝐶𝐻𝐷𝐹(𝑥) = - 𝑙𝑛(𝑅𝐶𝐷𝐹(𝑥))

𝐼𝐶𝐷𝐹(𝑥) = 𝐶𝐷𝐹^-1(𝑥)

𝑀𝐺𝐹(𝑡) = 𝐄[𝑒^𝑡𝑋]

∫_-∞^𝑥𝑓(𝑥)𝑑𝑥

1 - 𝑆(𝑥)

1 - 𝑒^{[ -∫_-∞𝑥𝘩(𝑥)𝑑𝑥 ]}

𝐹’(𝑥)

-𝑆’(𝑥)

𝘩(𝑥) 𝑒^{[ -∫_-∞𝑥𝘩(𝑥)𝑑𝑥 ]}

1 - 𝐹(𝑥)

∫_𝑥^∞𝑓(𝑥)𝑑𝑥

e ^{[ -∫_-∞𝑥𝘩(𝑥)𝑑𝑥 ]}

𝐹’(𝑥) / [1 - 𝐹(𝑥)]

𝑓(𝑥) / ∫_𝑥^∞𝑓(𝑥)𝑑𝑥

-𝑆’(𝑥) / 𝑆(𝑥)