Document Sample

Probability and Reliability Outline: How do I describe a random variable? Basic descriptive statistics Basic probability concepts Probability distributions How do I calculate Binomial distributions probabilities? Exponential distributions How do I calculate Reliability reliability Confidence intervals Basic descriptive statistics Mean Average (expected) value Median 50th-percentile value (50% above, 50% below) Mode Most likely value Standard deviation Measure of dispersion about the mean Coefficient of variation Dimensionless measure of variability (COV=s/m) Higher-order moments Skewness, flatness, etc. Basic probability concepts Three (3) basic axioms Axioms of probability Discrete and continuous distributions PDF’s and CDF’s Probability calculations “What is the probability of …?” Statistical independence (important concept) A B IF A and B are independent A “intersects”B P(A B) = P(A)·P(B/A) = P(A) P(B) Sample probability functions (density, and cumulative distribution functions) a F X (a ) P[ X a ] f X ( x)dx Some common distributions Discrete Continuous Binomial Uniform Pascal (r) Normal Geometric (r=1) Lognormal Poisson Gamma (r) Negative Binomial Exponential (r=1) Probability calculations What is the probability of a random variable being equal to some value ? What is the probability of being less than some value? a F X (a ) P[ X a ] f X ( x)dx What is the probability of being between two values? b P[a X b] f X ( x)dx FX (b) FX (a) a Binomial distribution Binomial distribution is a discrete distribution, and is used for situations where individual trials, experiments, or tests have only two possible outcomes such as success/failure, go/no-go, defectetive/non-defective or pass/fail. N! p( r / N , p ) p (1 p ) N r r r!( N r )! N is the sample size ( in a trial) r is the number of events of interest observed in the sample (i.e. failures, or defectives) p is the proportion of events of interest in the population p(r / N, p) also can be written as P (r, N, p) not to be confused with p. Binomial Distribution (cont’d) Example 1: The proportion of defectives in a product line is 1%. Take a random sample of 25. What is the probability that this sample has one defective ? N = 25, p = 0.01, r = 1 N! p(r 1 / N 25, p 0.01) p (1 p ) N r .1964 r r!( N r )! What if p = 0.001 ? N! p(r 1 / N 25, p 0.001 ) p (1 p ) N r .0244 r r!( N r )! Binomial Distribution (cont’d) Example 2: The failure rate of a device on demand is 10-4. What is the probability that 3 out of 4 devices fail at demand ? N = 4, r = 3, p = 10-4 N! p(r 3 / N 4, p 10 4 ) p (1 p ) N r 3.996 10 12 ~ Np r r!( N r )! Exponential Distribution The Exponential Distribution is widely used in reliability engineering to estimate failure probability as function of time. F ( , T ) P(T T / ) 0 0 T0 p' (T , ) exp(T ) p '(T , )dT 1 exp(T0 ) 0 is the failure rate T is time to fail 0.1 0.09 0.08 0.07 0.06 P(,T) 0.05 0.04 0.03 0.02 0.01 0 0 T Exponential Distribution (cont’d) Example 1: An electric motor's constant failure rate is 0.0004 failures/hr. Calculate probability of failure for a 150 hr mission. Operating time starts at t = 0. = 0.0004, T0 = 150 hr F ( , T0 ) P(T T0 / ) 1 exp(T0 ) 0.0582 Mean life expectancy E(T) = 1/ = 2500 hr What if = 0.004 ? F(,T0) = 0.4512, E(T) = 1/ = 250 hr Exponential Distribution (cont’d) Example 2: A CD player has an average record of successfully operating and providing listening enjoyment for more than 5,000 hours on the average before requiring repairs. A customer is planning on buying a CD player for installation in a boat that will be taking an extended cruise that will demand 4,000 hours of play before being able to obtain repairs or routine maintenance. How reliable will the CD player be for the customer? (Assume an exponential distribution) Exponential Distribution (cont’d) Mean life expectancy E(T) = 1/ = 5000 hr, so = 1/5000 = 0.00002 per hr F ( , T0 4000) P(T T0 4000 / ) 1 exp(T0 ) 0.5507 R(T0 4000 / ) 1 F ( , T0 4000) 0.4493 Reliability R(t) is the probability that no system failure will occur in a given time interval. As expected this is not a good reliability for such a task. Reliability as a function of System Complexity # of components Component Component in Series Reliability = Reliability = 99.999% 99.99% 100 99.9 99.01 250 99.75 97.53 500 99.50 95.12 1000 99.01 90.48 10,000 90.48 36.79 100,000 36.79 0.01 Confidence Intervals Example 1: For quality control, a random sample of 25 are taken from a large population of products. What is the defective proportion, if one defective product is found in the sample ? N = 25, r = 1 p = ? pe = r/N = 1/25 = 0.04 Confidence Intervals (cont’d) Z2 pe (1 pe ) Z 2 From Binomial: N pe Z N Z 2 2N N 4N 2 Where pe is the observed p, and Z is the standard score for confidence asked For a 95% confidence interval, Z = 1.96 pU = 0.1954 pL = 0.0071 It means that if the same population is sampled on numerous occasions and interval estimates are made on each occasion, the resulting intervals would bracket the true population parameter in approximately 95% of the cases. N pU pL 25 0.1954 0.0071 100 0.0984 0.0157 500 0.0610 0.0260 Confidence Intervals (cont’d) Example 1: An electric motor fails after 150 hr operation. Estimate the 90% confidence interval of the failure rate . e= 1/150 = 0.0067 (failures per hour) Use exact confidence interval for Poisson process: re is the observed r=1 (one failure). is for 100(1-) % confidence ; that is = 0.1 re re e U T (U T ) r r 0 p(r / U , T ) r 0 r! /2 re 1 re 1 e LT (LT ) r r 0 p(r / L , T ) r 0 r! 1 / 2 U= 0.032, L= 0.000355 U < < L Questions / Discussion

DOCUMENT INFO

Shared By:

Categories:

Tags:
probability theory, possible outcomes, sample space, Conditional Probability, the experiment, probability distributions, random variables, even number, probability distribution, Introduction to Probability

Stats:

views: | 79 |

posted: | 5/16/2010 |

language: | English |

pages: | 20 |

OTHER DOCS BY pengxiang

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.