Neat and comprehensive cheat sheet of all the formulas that you need during the examination 1CV40 - Quality and Reliability Engineering. On the last page is still a bit of room for additions.
𝑛 Normal distribution:
Chapter 2 λ(𝑡) = ∑ λ𝑖 (𝑡)
𝑖=1 1 1 (𝑡−𝜇) 2 𝑡−𝜇
− ∙
𝑅(𝑡 ) = 𝑃 (𝑇 ≥ 𝑡) 𝑛 𝑓 (𝑡) = ∙ 𝑒 2 𝜎2 𝑧=
√2𝜋 ∙ 𝜎 𝜎
𝑅(𝑡 ) = ∏ 𝑅𝑖 (𝑡 )
𝐹 (𝑡) = 𝑃 (𝑇 < 𝑡) = 1 − 𝑅(𝑡) 𝑖 =1
𝑀𝑇𝑇𝐹 = 𝜇 𝑠𝑡. 𝑑 = 𝜎
− ∑𝑛 ( )
𝑖=1 λ𝑖 𝑡 ∙𝑡
𝑑𝐹 (𝑡) 𝑑𝑅 (𝑡 ) =𝑒
𝑓 (𝑡) = = − 𝐶𝐷𝐹 = 𝑃(𝑍 ≤ 𝑧) = Φ(𝑧) 𝑅(𝑡) = 1 − Φ
𝑑𝑡 𝑑𝑡 Poisson distribution: The chance that n fa ilures
Lognormal distribution (log(t) has a normal distribution):
𝑡′ in range t=0 to t.
𝐹 (𝑡) = ∫ 𝑓 (𝑡 ′ )𝑑𝑡′ 1 −
1
∙(ln(
𝑡
)) 2
0 𝑒 −𝜆∙𝑡 ∙ (𝜆 ∙ 𝑡)𝑛 𝑓 (𝑡) = ∙𝑒 2∙𝑠2 𝑡𝑚𝑒𝑑
𝑝𝑛 (𝑡) = √2𝜋 ∙ 𝑠 ∙ 𝑡
∞ 𝑛!
𝑅(𝑡 ) = ∫ 𝑓 (𝑡 ′ )𝑑𝑡′ 1 𝑡
𝑡′
𝑝0 (𝑡) = 𝑅(𝑡) = 𝑒 −𝜆∙𝑡 𝐹 (𝑡) = Φ ( ln( ))
𝑠 𝑡𝑚𝑒𝑑
𝑃 (𝑎 ≤ 𝑇 ≤ 𝑏) = 𝑅(𝑎 ) − 𝑅(𝑏) 𝐸[𝑛] = 𝜎 2 = 𝜆 ∙ 𝑡
1 𝑡
= 𝐹 (𝑏) − 𝐹(𝑎) Calculate the probability that there have been
𝑅 (𝑡 ) = 1 − Φ ( ln( ))
𝑠 𝑡𝑚𝑒𝑑
∞ n failures in time t=0 to t:
2 2
𝑀𝑇𝑇𝐹 = ∫ 𝑡 ∙ 𝑓(𝑡 )𝑑𝑡 𝑠 −𝜆∙𝑡 𝑛 𝜎 2 = 𝑡𝑚𝑒𝑑 2 ∙ 𝑒 𝑠 ∙ [𝑒 𝑠 − 1]
0 𝑒 ∙ (𝜆 ∙ 𝑡)
𝑅𝑠 (𝑡) = ∑ 2 2/2
∞ 𝑛=0 𝑛! 𝑡𝑚𝑜𝑑𝑒 = 𝑡𝑚𝑒𝑑/ 𝑒 𝑠 𝑀𝑇𝑇𝐹 = 𝑡𝑚𝑒𝑑/ 𝑒 𝑠
𝑀𝑇𝑇𝐹 = ∫ 𝑅 (𝑡) 𝑑𝑡 Gamma distribution:
0 The Gamma distribution is closely related to
the Poisson, and calculates the probability that
∞ 𝑡𝑟−1 ∙𝑒 −𝑡/𝛼
2 2
the kth failure will occur by time t (Yk = t of kth): 𝑓 (𝑡) = 𝑀𝑇𝑇𝐹 = 𝛼𝛾 𝜎 2 = 𝛾𝛼 2
𝜎 = ∫ 𝑡 ∙ 𝑓(𝑡 ) 𝑑𝑡 − (𝑀𝑇𝑇𝐹 )2 𝑡𝛾 ∙𝛤 (𝛾)
0 𝑘 (𝜆 ∙ 𝑡)𝑖 𝐼(𝑡/𝛼,𝛾) 𝐼 (𝑡/𝛼,𝛾)
𝐹𝑌 (𝑡) = 1 − 𝑒 −𝜆∙𝑡 ∙ ∑ 𝐹 (𝑡) = 𝑅(𝑡) = 1 −
∞ 2 𝑖 =0 𝑖! 𝛤(𝛾) 𝛤(𝛾)
= ∫0 (𝑡 − 𝑀𝑇𝑇𝐹 ) 𝑑𝑡
𝑡𝑚𝑜𝑑𝑒 = 𝛼(𝛾 − 1) 𝑓𝑜𝑟 𝛾 > 1
𝑡′ 𝐸[𝑌𝑘 ] = 𝑘/𝜆 𝜎 2 = 𝑘/𝜆2
− ∫0 𝜆 (𝑡′)𝑑𝑡′
𝑅(𝑡 ) = 𝑒
𝑘−1
𝑡′
𝑌𝑚𝑜𝑑𝑒 =
𝜆
Chapter 5
∫0 𝜆 (𝑡′)𝑑𝑡′
𝐴𝐹𝑅 = 𝑛
𝑡 If the system is guaranteed to work until t0: 𝑅𝑠 (𝑡)𝑠𝑒𝑟𝑖𝑒𝑠 = ∏ 𝑅𝑖 (𝑡)
𝑖=1
𝑅 (𝑇0 + 𝑡) 𝑅(𝑡 ) = 𝑒 −𝜆∙(𝑡−𝑡0)
𝑅(𝑡 | 𝑇0 ) = 𝑛
𝑅(𝑇0 ) 𝑅𝑠 (𝑡)𝑝𝑎𝑟 = 1 − ∏ (1 − 𝑅𝑖 (𝑡))
𝑑𝑅 (𝑡) 𝑖=1
∞ 𝑓 (𝑡) = − = 𝜆 ∙ 𝑒 −𝜆∙(𝑡−𝑡0 )
𝑑𝑡 1 1
𝑀𝑇𝑇𝐹 (𝑡 | 𝑇0 ) = ∫ 𝑅 (𝑡 | 𝑇0 )𝑑𝑡 𝑀𝑇𝑇𝐹𝑠𝑒𝑟𝑖𝑒𝑠 = ∑𝑛 =
1 𝑙𝑛0,5 𝑖=1 λ𝑖 (𝑡) ∑𝑛
𝑖=1 1/ 𝑀𝑇𝑇𝐹𝑖
0
1 ∞ 𝑀𝑇𝑇𝐹 = 𝑡0 + 𝑡𝑚𝑒𝑑 = 𝑡0 +
λ λ
= ∙ ∫𝑇 𝑅(𝑡′) 𝑑𝑡 1 1 1
𝑅 (𝑇0 ) 0
𝑀𝑇𝑇𝐹𝑝𝑎𝑟 = + −
Chapter 4 𝜆1 𝜆2 𝜆1 + 𝜆2
Chapter 3 High and low-level redundancy indicate how a system is
Weibull distribution: designed. The variable m indicates the number of horizontal
Exponential distribution (CFR with λ(t)=λ): components and n indicates the number of vertical
used to calculate the chance that the next 𝛽 𝑡 𝛽−1 components:
failure will happen in range t=0 to t. 𝜆 (𝑡) = ∙( )
𝜃 𝜃
𝑅(𝑡 ) = 𝑒 −𝜆∙𝑡 𝑡′ 𝑡 𝛽
∫0 𝜆 (𝑡′)𝑑𝑡′ −( )
𝑅(𝑡 ) = 𝑒 − = 𝑒 𝜃
𝐹 (𝑡) = 1 − 𝑒 −𝜆∙𝑡
𝛽 𝑡 𝛽−1 −( 𝑡 ) 𝛽
𝑑𝐹 (𝑡) −𝜆∙𝑡 𝑓 (𝑡) = ∙( ) ∙𝑒 𝜃
𝑓 (𝑡) = = 𝜆 ∙𝑒 𝜃 𝜃 The chance that x=k out of n identical components work is:
𝑑𝑡
𝑀𝑇𝑇𝐹 = 𝜃 ∙ Γ(1 + 1/𝛽) 𝑛
1 1 𝑃(𝑥 ) = ( ) 𝑅𝑥 (1 − 𝑅)𝑛−𝑥
𝑀𝑇𝑇𝐹 = 𝜎 = 𝜎2 = 𝑥
λ λ2
2 1 2
1 𝜎 2 = 𝜃2 ∙ [Γ (1 + ) − [Γ (1 + )] ] 𝑛!
𝛽 𝛽 = 𝑅𝑥 (1 − 𝑅)𝑛−𝑥
𝑡𝑚𝑒𝑑 = ∙ ln(0,5) 𝑥! (𝑛 − 𝑥 )!
λ 1/𝛽
−𝜆∙𝑡
𝑡𝑅 = 𝜃 ∙ (− ln(𝑅) ) 𝑛
𝑅(𝑡 | 𝑇0 ) = 𝑅(𝑡) = 𝑒 𝑅𝑠 = ∑ 𝑃(𝑥)
1/𝛽 𝑥=𝑘
𝑡′ 𝑡𝑚𝑜𝑑𝑒 = 𝜃 ∙ (1 − 1/𝛽) 𝛽> 1
𝜆 𝑖 (𝑡′ )𝑑𝑡′
𝑅𝑖 (𝑡 ) = 𝑒 − ∫0 ∞
𝑀𝑇𝑇𝐹 = ∫0 𝑅𝑠 (𝑡) 𝑑𝑡 = ∙ ∑𝑛𝑥=𝑘
1 1
1 𝛽 𝜆 𝑥
λ(𝑡) = 𝛽 ∙ 𝑡𝛽−1 ∙ ∑𝑛𝑖=1 ( )
𝜃𝑖
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller marieketbk. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $3.75. You're not tied to anything after your purchase.
