Unformatted text preview: Engr 9397 – Week 4 Probability Distribu9ons Engineering Probability Models • Probability es9mates can be obtained either through experimenta9on or by a theore9cal model • Example: number of 9mes tails appears when a coin is tossed 3 9mes. • We could conduct and experiment where we would toss 3 coins 1000 9mes and observe the results are many tests available OR • We could assume coins have an unbiased probability of 0.5 and reason probability would be P(X=3) = p3 = 0.53 = 0.125 Probability Models • Probability es9mates can be obtained either through experimenta9on or by a theore9cal model • Example: number of 9mes tails appears when a coin is tossed 3 9mes. • We could conduct and experiment where we would toss a 3 coins 1000 9mes and observe the results are many tests available OR • We could assume coins have an unbiased probability of 0.5 and reason probability would be P(X=3) = p3 = 0.53 = 0.125 Types of Probability Distribu9ons • There are over 500 probability distribu9ons! • Probability distribu9ons most commonly used in engineering applica9ons include: • Discrete Models – Bernoulli sequence, Binomial, Poisson • Con9nuous Models – Uniform, Normal, Lognormal, Exponen9al Probability Distribu9on Models • Models are conceptual and are deﬁned by parameters (usually 3 or less) using es9mates from the data • We are oZen using point es9mates only, which are subject to variability, so we need to check to see whether the model ﬁts the data • To check how well a given model matches with the data, goodness of ﬁt tests are used Probability distribu9on func9on condi9ons • All probability models follow several rules • In order for any func9on, f(x) to represent a probability distribu9on, the following condi9ons must be sa9sﬁed: • Only the equa9on and shape of the distribu9on curve is diﬀerent, and the choice of appropriate distribu9on depends on the data – Each value of f(x) must lie between 0 and 1, inclusive (has nonnega9ve densi9es or masses) – The area under the probability distribu9on must equal 1 (sum of all values of f(x) must equal 1) Discrete Probability Distribu9ons • Probability Distribu9on Func9on – arguments (x) are values of a random variable func9onal values f(x) are probabili9es x F(x) 1 .15 2 .25 3 .3 4 .15 5 .1 6 .05 • Mean: µ = • Variance: σ2 = • Recall that a random variable can be either con9nuous or discrete Bernoulli sequence and the Binomial distribu9on • Many problems in engineering can be modeled as just 2 possible outcomes in a trial. Ex. yes or no, success or failure, head or tail • Problems of this type may be modeled by a Bernoulli sequence • The probability of x occurrences in n trials of a Bernoulli sequence is given by the Binomial distribu9on, a discrete probability distribu9on that has two parameters, n (# of trials) and p (probability of success) Binomial Distribu9on • Binomial distribu9ons are used when: – Experiment consists of n trials/repe99ons – Each trial has two mutually exclusive outcomes – The probability of occurrence (p) or non
occurrence (q=1
p) of the event in each trial is constant – Trials are independent of each other Use of Binomial model • In spite of its simplicity, the Binomial model is quite useful in many engineering applica9ons • It will work as long as we can determine n and p, and assume individual, discrete trials • For space
9me problems (# of telephone calls/ hour, # cracks/m^2) we would use the Poisson distribu9on • The binomial distribu9on is used oZen in analyzing data about defects, and is the basis for process control charts that we will study later Binomial Distribu9on • Binomial Distribu9on: Probability of x successes (or n
x failures) out of n independent trials – x is a random variable value – Probability of success on a trial denoted as ‘p’ – n and p are the 2 parameters of this distribu9on • Probability of x consecu9ve successes is px • Probability of complementary outcome (failure) is 1
p Binomial Distribu9on con9nued the cumula9ve binomial distribu9on (shown above) is useful when we want to know the probability of:
x successes at least x success more than x successes For the binomial distribu9on, the mean and standard devia9on simplify to: Binomial Distribu9on
Examples • A container is ﬁlled with 30 packages to be shipped to two diﬀerent des9na9ons. Can the binomial distribu9on be used to ﬁnd the probability that 10 of the ﬁrst 20 packages removed from the container will go to one of the des9na9ons? Why or why not? Binomial Distribu9on
Examples 1. What is the probability that 15 of 16 lawn mowers will start on the ﬁrst pull if the manufacturer’s claim is true that ﬁrst
pull star9ng can be expected 90% of the 9me? 2. Find the mean and the standard devia9on of the distribu9on of the number of ‘shorted’ transistors in a lot of 100 transistors, if the percentage of shorted transistors produced is 15%. Probability of Acceptance & the Binomial Distribu9on • For manufactured products, some9mes the en9re lot is rejected if number of samples inspected exceeds an ‘acceptance number’ • OZen we’re looking for the probability that a lot will be accepted • Because the lot is large compared to n, it’s ok to assume sampling without replacement and ~equal probabili9es for each trial • Using the binomial distribu9on, we can determine the probability of acceptance: L( p) = B(c;n, p)
€ Other Distribu9ons with similar proper9es to the Binomial • Geometric Distribu9on: same assump9ons (2 possible outcomes, independent sampling) – Used to calculate the probability that the ﬁrst success occurs on trial (x) • Mul9nomial distribu9on: used to obtain x1 outomces of type 1, x2 outcomes of type 2.. where possible outcomes have probabili9es p1+ p2+ …= 1 Sampling Inspec9on • Some9mes, an en9re lot is rejected if number of samples inspected exceeds an ‘acceptance number’ – A proper sampling plan speciﬁes samples size & limits • Inspec9on gets trickier when you introduce a prototype environment • Virtual inspec9ons/reviews, correct market
driven speciﬁca9ons, and an eﬀec9ve design
build transi9on help catch mistakes before prototypes are built. • Improvements in design eﬃciency also help shrink design cycles, reduce re
work, helping the bopom line by cuqng product development 9me and cost. Poisson Distribu9on • Useful approxima9on of the binomial distribu9on when n is large and p is small • One parameter is used to described the model λ called the average rate of occurrence, or mean occurrence rate – For Binomial, n must be known, whereas for the Poisson, n is unknown. • Here, np is ﬁxed at λ (both mean and variance =λ) Poisson Distribu9on con9nued • Mean & variance: • Poisson distribu9on • Cumula9ve Poisson distribu9on • Probability that rv X takes on a speciﬁc value x: λx − λ f ( x; λ ) − − > P ( X = x ) = e x! € Cumula9ve Poisson Distribu9on • The probability that a random variable (a.k.a. rv) X having the Poisson distribu9on with parameter λ assumes the value x is given by: P(X=x) = F(x; λ) – F(x
1; λ) • The probability that a rv having the Poisson distribu9on with paramter λ assumes a value greater than x is given by: P(X>x) = 1
F(x; λ) Example Poisson Distribu9on • If the propor9on defec9ve on inspec9on of stampings for an automo9ve quarter panel is 0.4%, – what is the probability of obtaining 3 or more defec9ve quarter panels on a day when 350 panels are stamped? – What are the mean and the standard devia9on of the number of defec9ve panels? Con9nuous Probability Distribu9ons • Most data measured in quality engineering applica9ons are con9nuous • Unlike discrete rvs, con9nuous rvs cannot directly be assigned a probability value • Instead of a probability histogram with individual rectangular bars represen9ng values and probabili9es, we now have a con9nuous func9on f(x) with base width Δx that approaches zero as the rv assumes an exact value • For con9nuous rv’s, f(x) is called a probability density func9on instead of a probability distribu9on Con9nuous probability Distribu9ons • The mean or expected value of a con9nuous probability distribu9ons is a func9on of a rv, ex. g(x), where: • And: Uniform Distribu9on – Equal probability that con9nuous rv assumes value x between α and β [0 and 1] – The probability that rv with a uniform distribu9on will lie between a and b where α < a < b < β is given by: Where and Normal Distribu9on • Most commonly encountered distribu9on • Symmetrical bell shape : many measurements (ex. length, diameter) follow this distribu9on • Normal distribu9on is described by 2 parameters, the average and the standard devia9on • Representa9ve of actual measurements • Convenient for assump9ons • Normal Distribu9on: Normal Distribu9on: Central Limit Theorem • Under very general condi9ons the distribu9on of the sum of n random variables approaches a normal distribu9on when n is large regardless of the shape of the contribu9ng variables • CLT forms the condi9ons for a normal distribu9on: – A large number of causa9ve factors aﬀec9ng the rv – Each factor has only a small contribu9on to the outcome – Each factor should be independent Central Limit Theorem • Allows for large samples (n > 30) to be generalized, allowing their characteris9cs to represent those of the en9re popula9on • If x is the mean of a random sample of size n taken from a popula9on with mean µ and standard devia9on σ, then the standardized mean, z, is the value of a rv whose distribu9on approaches the standard normal distribu9on as n ∞. • Said another way, if x is a normally distributed rv with mean µ and standard devia9on σ, the corresponding standardized rv z is also normally distributed with mean 0 and std dev 1 • Standard normal distribu9on: N (z ; 0, 1) where z is: Why z? • The pdf and the cdf of a normal distribu9on cannot be integrated directly, we have to rely on tables to obtain the cumula9ve probabili9es for a given set of parameters • We’d need an inﬁnite number of tables because of the inﬁnite possibili9es! • We can standardize the value by a transforma9on so that only one value, z, is used as the table look
up value Normal Distribu9on Proper9es • The normal distribu9on has 2 parameters: µ, σ • The sample mean and sample standard devia9on are es9mates of the 2 parameters • A normal distribu9on is symmetric about the mean, has zero skew, and the mean=median=mode • A normal curve can be drawn using these coordinates [x, n(x;µ, σ)] and can take on many shapes • Probabili9es of normally distributed rv’s are found by ﬁnding examining the area(s) under the normal curve Standard Normal Distribu9on • Using tables that give values of the cumula9ve standard normal distribu9on, N(x;0,1), probabili9es for normal distribu9ons can be calculated • These sta9s9cal tables give leZ
hand tail areas which represents the probability that std normal rv assumes value less than x • To compute probabili9es, standardize the variable (s) (ﬁnd z), draw a sketch indica9ng shading area of interest and ﬁnd the corresponding probability from the tables • For nega9ve values of z, Summary: Standard Normal Distribu9on • To facilitate calcula9on of normal probabili9es, we approximate results using the distribu9on of a standardized random variable (z) having value: • If x is a normally distributed rv with mean µ and standard devia9on σ, the corresponding standardized rv is also normally distributed, with: Normal distribu9on
calcula9ng probabili9es • Find the probability that a rv having a normal distribu9on with mean
2.5 and standard devia9on 1.6 will assume a value between
3 and 1. • Speciﬁca9ons require that the thickness of asphalt on a road be at least 3 inches, with a standard devia9on of no more than 0.15 inch. Believing that there is a quality problem involving the paving contractor, an engineer takes a single core sample, obtaining a thickness of 2.5 inches. Is the engineer’s suspicion jus9ﬁed? Checking for Normality • If the data distribu9on is not symmetric and bell
shaped, it cannot be considered normal • How do check to see if the sample data we collect is normal, lognormal..? • Several ploqng methods, including : the normal
probability plot, the normal scores plot can help to determine if the data are normally distributed • One can also make transforma9ons (logarithmic, spare
root, reciprocal) to the data and then subsequently check for normality Normal Probability Plots • Good for checking to see if data comes from a bell
shaped or normal distribu9on • The assump9on of a normal distribu9on, where a data popula9on can be described by its mean and its standard devia9on, is a powerful once used oZen in quality engineering applica9ons • Ensuring that the data is in fact normal will prevent faulty conclusions about the data characteris9cs Normal Distribu9ons – Applica9on to Quality Management • The assump9on of normality is oZen made about data measured in order to analyze and monitor quality
related concerns • Knowing µ and σ tells us what values the data will assume and their associated probabili9es • Ex. probability that rv x will take on values within k standard devia9ons from its mean: k Prob 1 .68 2 .95 3 .997 Normal approxima9ons for discrete distribu9ons • When n is large, the Binomial and Poisson distribu9ons can be approximated by the Normal distribu9on Lognormal Distribu9ons • A non
nega9ve rv X has a logarithmic distribu9on whenever Y=log(X) has a normal distribu9on • The use of the lognormal distribu9on is very similar to the normal, the only diﬀerence is that the logs of the variables Y=log(X) are normally distributed instead of the original X • we use: Z = (log(X)
µy)/σy • We can also take the natural logs of the data and treat the data as a normal distribu9on Lognormal Distribu9ons – The probability that a rv with a lognormal distribu9on (with mean µ and standard devia9on σ) will assume a value between a and b is equivalent to: – the probability that a normally distributed rv (with mean α and standard devia9on β) will assume a value between ln a and ln b – µ and σ can be expressed by the lognormal distribu9on’s parameters: 2 µ = eα + β / 2 and Lognormal Distribu9ons • The probability of a rv having a normal distribu9on with mean µ and standard devia9on σ assuming a value between x1 and x2 is equivalent to: • The probability of a rv having a lognormal distribu9on, with the same mean and standard devia9on, assuming a value between ln x1 and ln x2. • To determine the standardized values of a lognormal distribu9on, replace x1 and x2 with ln x1 and ln x2 and then calculate z1 and z2 . Then determine the probability of interest using the standard normal tables. Lognormal Distribu9on Example • If experience shows that the natural logarithms of processing 9mes for jobs submiped to a computer have the normal distribu9on with a mean of 1.5 and a standard devia9on of 0.4, ﬁnd the probability that such a 9me will lie between 3 and 6 minutes. Exponen9al distribu9on • Important in reliability theory and queuing theory for and • Knowing either the mean, standard devia9on or alpha allows one to calculate all three variables as well as the associated probability ...
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 Winter '11
 SusanHunt
 #, 15%, 0.4%, the00, Engineer00, lognormal..

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