CGE676 Chapter 2: Statistical Reliability PDF

CGE676 Reliability and Maintenance Engineering Chapter 2 - Statistical Reliability Rules of Probability Basic Statistic in Reliability Probability E.g Flip a coin 1 time Examples Eith...

CGE676 Reliability and Maintenance Engineering Chapter 2 - Statistical Reliability Rules of Probability Basic Statistic in Reliability Probability E.g Flip a coin 1 time Examples Either H or T ❖Toss a fair coin P(A) = Head – 1 P(B) = Tail – 1 ❖Roll a dice Sample space = H,T E.g Roll a dice 1 time Either 1,2,3,4,5,6 P(A) = Get no 1 P(B) = Get no 2 Sample space = 1,2,3,4,5,6 Faculty of Chemical Engineering Basic Statistic in Reliability Probability For any event A: 0≤P(A)≤1 P(S) = 1 if events A, B mutually exclusive P(A U B) = P(A) + P(B) Complement rule: P(A’)=1-P(A) Faculty of Chemical Engineering Basic Statistic in Reliability Probability Conditional probability P(A) = P(A|S) P(A), P(B) and P(A∩B) are “absolute” terms with respect to S P(B|A) is “relative” to the reduced sample space A Faculty of Chemical Engineering Basic Statistic in Reliability Probability Conditional probability P(B|A) = ? P(B|A)  P(A|B) P(A|B) ≥ P(A ∩ B) P(B|A) ≥ P(A ∩ B) Example: Someone tosses a dice, covers it up and tells you that the number is greater than 3. What is the probability that the number is even? Faculty of Chemical Engineering Basic Statistic in Reliability Probability Conditional probability Solution: A = {the number is greater than 3) B = {the number is even} P(B|A)= P(A∩B) = P({4,6}) = 2 P(A) P({4,5,6}) 3 Faculty of Chemical Engineering Basic Statistic in Reliability Probability Multiplicative Rule P(A∩B) = P(A|B)P(B) = P(B|A)P(A) ✓ If P(B|A) = P(B), P(A|B) = P(A) P(A∩B) = P(A)P(B) ✓ Events A and B are independent if and only if P(A∩B) = P(A)P(B) Example: Suppose 5 defective light bulbs were inadvertently packed in a box with 20 good ones. Someone randomly selected 2 light bulbs from the box (without replacement). What is the probability that both of them are defective? Faculty of Chemical Engineering Basic Statistic in Reliability Probability Multiplicative Rule Solution: A = {first light bulb removed is defective) B = {second light bulb removed is defective} P (A) = 5/25 = 1/5 P(B|A)= 4/24 = 1/6 P(A∩B) = P(B|A)P(A) = (1/6)(1/5) = 1/30 Faculty of Chemical Engineering Basic Statistic in Reliability Probability Law of Total Probability (LTP) If A1, A2, ….. An are mutually exclusive and exhaustive event in S, then for any other event B in sample space S, P(B) = P(B|A1)P(A1)+……+P(B|An) = P(B|Ai)P(Ai) Faculty of Chemical Engineering Basic Statistic in Reliability Probability Law of Total Probability (LTP) Example: An electrical appliance retailer sells three brands of TVs, and their market shares are respectively 50%, 30%, 20%. It is estimated that 2.5% of brand 1’s, 2% of brand 2’s, and 1% of brand 3’s TV will be returned by customers for repair while under the one-year warranty offered by manufacturers. i) What is the probability that a random customer purchases a brand 1’s TV and has to return it to the retailer for repair covered by the one-year warranty? ii) What is the probability that a TV sold by the retailer will be returned for warranty repair work? Faculty of Chemical Engineering Basic Statistic in Reliability Probability Law of Total Probability (LTP) Solutions: Ai = {TV sold is brand i}, i=1,2,3 B = {TV requires warranty repair work} P(A1) = 0.5, P(A2) = 0.3, P(A3) = 0.2 P(B|A1)=0.025, P(B|A2)=0.02, P(B|A3)=0.01 i) P(B∩A1)= P(B|A1) P(A1)=0.0125 ii) P(B)= P(B|A1) P(A1) + P(B|A2) P(A2) +P(B|A3) P(A3) =0.0125 + 0.02x0.3 + 0.01x0.2 =0.0205 Faculty of Chemical Engineering Basic Statistic in Reliability Probability of failure, F(t) A variable is a quantity whose value changes Random variables – is a set of possible values from a random experiment. Faculty of Chemical Engineering Probability distribution Classified into discrete and continuous – Discrete: involve the counting of particular event (eg: rolling dice, drilling exploratory wells in a new field or basin – Continuous: Eg. Petroleum reserve, porosity, permeability, recovery factor, cash flow, etc. Faculty of Chemical Engineering Basic Statistic in Reliability Discrete random variables - Can assume a finite number of values or a countable infinity of values or a variable whose value is obtained by counting. Example 1: Toss a coin twice, count Tails? Faculty of Chemical Engineering Basic Statistic in Reliability Continuous random variables - set of possible values is an entire interval of numbers/ variable whose value is obtained by measuring. Faculty of Chemical Engineering Probability distribution Cumulative distribution function (CDF) F(x) = P(X≤x) = ∑v:v0), i.e., each half is mirrored by the other ▪ Asymptotic to the horizontal axis (approach 0 as x goes to ∞ and -∞ ▪ Unimodal (maximum occurs at x= µ) ▪ A family of curves (for different µ and σ) ▪ Area under the curve = 1 Faculty of Chemical Engineering Normal distribution distribution Application in reliability engineering: Can be used to describe the lifetime behavior of component suffering from wear-out mechanisms. This becomes evident from the hazard rate function. Faculty of Chemical Engineering Lognormal distribution A random variable X is said to have the lognormal distribution, with parameters μ and σ, if ln(X) has the normal distribution with mean μ and standard deviation σ. The lognormal density function is given by: Properties of lognormal distribution: Having the close relationship with normal distribution Takes only positive values Skewed to the right Approached when σ gets smaller Faculty of Chemical Engineering Lognormal distribution Application in reliability engineering: Suited for fitting to lifetime variables that are governed by fatigue processes. Can be used to describe the lifetime behavior of component suffering from wear-out mechanisms. Evident from the hazard rate function. Faculty of Chemical Engineering Exponential distribution A widely used probability distribution function in maintenance and reliability work Probability density function Probability of failure, distribution function Reliability f (x) Exponential x Faculty of Chemical Engineering Probability density function (PDF) Properties of PDF: Faculty of Chemical Engineering Probability density function (PDF) Example 3: Probability density function for r.v X, Calculate P(-10≤x≤10)? Faculty of Chemical Engineering Probability density function (PDF) Solution Faculty of Chemical Engineering Cumulative density function (CDF) The cumulative distribution function (CDF) F(x) of a continuous rv X with pdf fX(x) is given by : Faculty of Chemical Engineering Cumulative density function (CDF) Properties of CDF: For a continuous rv X whose pdf is defined by fX(x), its mean is given by Its variance, var(X) or σX2 is given by Faculty of Chemical Engineering Relationship between measures Faculty of Chemical Engineering Applications in reliability engineering problems Example 4: Assume that the time to failures of an equipment is exponentially distributed and its failure rate is 0.003 failures per hour. Calculate the equipment's reliability for a 10-hour mission? Faculty of Chemical Engineering Application in reliability problems Solution: An approximately 97% chance that the equipment will not fail during the 10-hour mission. More specifically, its reliability will be 0.9704. Faculty of Chemical Engineering Laplace transform Provides a useful method of solving certain types of differential equations when certain initial conditions are given, especially when the initial values are zero. Laplace transform of function f(t) is defined by; Faculty of Chemical Engineering Laplace transform Laplace Transforms of some frequently occurring functions in maintainability, maintenance, and reliability studies: f(t) f(s) c (a constant) tK, for K=0, 1, 2, 3,… Faculty of Chemical Engineering Next Lecture…… Reliability network Faculty of Chemical Engineering