\(P(x < k) = (\text{base})(\text{height}) = (k0)\left(\frac{1}{15}\right)\) In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. for 8 < x < 23, P(x > 12|x > 8) = (23 12) ) What is the 90th percentile of square footage for homes? In statistics, uniform distribution is a term used to describe a form of probability distribution where every possible outcome has an equal likelihood of happening. 12= Example The data in the table below are 55 smiling times, in seconds, of an eight-week-old baby. 2 c. Ninety percent of the time, the time a person must wait falls below what value? I'd love to hear an explanation for these answers when you get one, because they don't make any sense to me. a+b The Uniform Distribution by OpenStaxCollege is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted. State this in a probability question, similarly to parts g and h, draw the picture, and find the probability. so f(x) = 0.4, P(x > 2) = (base)(height) = (4 2)(0.4) = 0.8, b. P(x < 3) = (base)(height) = (3 1.5)(0.4) = 0.6. However, if another die is added and they are both thrown, the distribution that results is no longer uniform because the probability of the sums is not equal. Correct me if I am wrong here, but shouldn't it just be P(A) + P(B)? It is defined by two different parameters, x and y, where x = the minimum value and y = the maximum value. The probability density function of X is \(f\left(x\right)=\frac{1}{b-a}\) for a x b. The data follow a uniform distribution where all values between and including zero and 14 are equally likely. Uniform Distribution. The data that follow record the total weight, to the nearest pound, of fish caught by passengers on 35 different charter fishing boats on one summer day. If a person arrives at the bus stop at a random time, how long will he or she have to wait before the next bus arrives? 23 2 Find the 30th percentile for the waiting times (in minutes). 14.42 C. 9.6318 D. 10.678 E. 11.34 Question 10 of 20 1.0/ 1.0 Points The waiting time for a bus has a uniform distribution between 2 and 11 minutes. Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. 1), travelers have different characteristics: trip length l L, desired arrival time, t a T a, and scheduling preferences c, c, and c associated to their socioeconomic class c C.The capital and curly letter . \(0.3 = (k 1.5) (0.4)\); Solve to find \(k\): The probability of waiting more than seven minutes given a person has waited more than four minutes is? A random number generator picks a number from one to nine in a uniform manner. Let X = the number of minutes a person must wait for a bus. The answer for 1) is 5/8 and 2) is 1/3. 12 The possible values would be 1, 2, 3, 4, 5, or 6. 1 a+b The probability is constant since each variable has equal chances of being the outcome. \(X\) = The age (in years) of cars in the staff parking lot. The cumulative distribution function of \(X\) is \(P(X \leq x) = \frac{x-a}{b-a}\). Then \(X \sim U(0.5, 4)\). If we create a density plot to visualize the uniform distribution, it would look like the following plot: Every value between the lower bounda and upper boundb is equally likely to occur and any value outside of those bounds has a probability of zero. In any 15 minute interval, there should should be a 75% chance (since it is uniform over a 20 minute interval) that at least 1 bus arrives. 230 P(x>2) If the waiting time (in minutes) at each stop has a uniform distribution with A = 0 and B = 5, then it can be shown that the total waiting time Y has the pdf $$ f(y)=\left\{\begin{array}{cc} \frac . a. The probability that a randomly selected nine-year old child eats a donut in at least two minutes is _______. P(x>2) Find the probability that a randomly chosen car in the lot was less than four years old. 4 The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. What is the probability that a person waits fewer than 12.5 minutes? \nonumber\]. Let X = the time, in minutes, it takes a student to finish a quiz. 12 1 \(f(x) = \frac{1}{15-0} = \frac{1}{15}\) for \(0 \leq x \leq 15\). a. = citation tool such as. On the average, how long must a person wait? =0.8= The waiting time at a bus stop is uniformly distributed between 1 and 12 minute. \(k = (0.90)(15) = 13.5\) The waiting time at a bus stop is uniformly distributed between 1 and 12 minute. It would not be described as uniform probability. The graph illustrates the new sample space. What is the probability density function? \(P(2 < x < 18) = 0.8\); 90th percentile \(= 18\). The total duration of baseball games in the major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive. P(17 < X < 19) = (19-17) / (25-15) = 2/10 = 0.2. In this distribution, outcomes are equally likely. 3.375 hours is the 75th percentile of furnace repair times. Creative Commons Attribution License percentile of this distribution? The distribution can be written as \(X \sim U(1.5, 4.5)\). \(f(x) = \frac{1}{9}\) where \(x\) is between 0.5 and 9.5, inclusive. Uniform Distribution between 1.5 and four with shaded area between two and four representing the probability that the repair time, Uniform Distribution between 1.5 and four with shaded area between 1.5 and three representing the probability that the repair time. The waiting time at a bus stop is uniformly distributed between 1 and 12 minute. c. Find the 90th percentile. ( This module describes the properties of the Uniform Distribution which describes a set of data for which all aluesv have an equal probabilit.y Example 1 . Note that the length of the base of the rectangle . (a) The probability density function of X is. for 0 x 15. Find the probability that a person is born at the exact moment week 19 starts. Uniform distribution refers to the type of distribution that depicts uniformity. The goal is to maximize the probability of choosing the draw that corresponds to the maximum of the sample. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. are not subject to the Creative Commons license and may not be reproduced without the prior and express written Find the value \(k\) such that \(P(x < k) = 0.75\). (15-0)2 The second question has a conditional probability. Find the probability that a bus will come within the next 10 minutes. Discrete uniform distributions have a finite number of outcomes. It means that the value of x is just as likely to be any number between 1.5 and 4.5. You already know the baby smiled more than eight seconds. = According to a study by Dr. John McDougall of his live-in weight loss program at St. Helena Hospital, the people who follow his program lose between six and 15 pounds a month until they approach trim body weight. 1 Get started with our course today. 0+23 The probability a person waits less than 12.5 minutes is 0.8333. b. Random sampling because that method depends on population members having equal chances. What does this mean? P(x 2) = (\text{base})(\text{height}) = (4 2)(0.4) = 0.8\), b. 23 = Discrete and continuous are two forms of such distribution observed based on the type of outcome expected. The notation for the uniform distribution is. c. Find the 90th percentile. 15 The probability that a randomly selected nine-year old child eats a donut in at least two minutes is _______. \(X \sim U(0, 15)\). 2.5 Legal. There is a correspondence between area and probability, so probabilities can be found by identifying the corresponding areas in the graph using this formula for the area of a rectangle: . The sample mean = 7.9 and the sample standard deviation = 4.33. P(x>12) Use Uniform Distribution from 0 to 5 minutes. We recommend using a Refer to Example 5.3.1. What percentage of 20 minutes is 5 minutes?). Buses run every 30 minutes without fail, hence the next bus will come any time during the next 30 minutes with evenly distributed probability (a uniform distribution). P(x < k) = (base)(height) = (k 1.5)(0.4), 0.75 = k 1.5, obtained by dividing both sides by 0.4, k = 2.25 , obtained by adding 1.5 to both sides. One of the most important applications of the uniform distribution is in the generation of random numbers. Theres only 5 minutes left before 10:20. Answer: a. 2 1 = Your probability of having to wait any number of minutes in that interval is the same. So, P(x > 12|x > 8) = \(\frac{\left(x>12\text{AND}x>8\right)}{P\left(x>8\right)}=\frac{P\left(x>12\right)}{P\left(x>8\right)}=\frac{\frac{11}{23}}{\frac{15}{23}}=\frac{11}{15}\). Find the mean, , and the standard deviation, . b. = \(\frac{a\text{}+\text{}b}{2}\) Find the probability that the commuter waits between three and four minutes. The number of miles driven by a truck driver falls between 300 and 700, and follows a uniform distribution. The probability \(P(c < X < d)\) may be found by computing the area under \(f(x)\), between \(c\) and \(d\). The sample mean = 11.49 and the sample standard deviation = 6.23. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. To keep advancing your career, the additional CFI resources below will be useful: A free, comprehensive best practices guide to advance your financial modeling skills, Get Certified for Business Intelligence (BIDA). Find the probability that a randomly selected furnace repair requires more than two hours. 15 2 k = 2.25 , obtained by adding 1.5 to both sides Introduction to Statistics is our premier online video course that teaches you all of the topics covered in introductory statistics. = Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. Answer: (Round to two decimal places.) The data that follow are the number of passengers on 35 different charter fishing boats. 2 Can you take it from here? What is the probability that the waiting time for this bus is less than 5.5 minutes on a given day? . Find the probability that the truck drivers goes between 400 and 650 miles in a day. \(X =\) __________________. a. 12, For this problem, the theoretical mean and standard deviation are. Develop analytical superpowers by learning how to use programming and data analytics tools such as VBA, Python, Tableau, Power BI, Power Query, and more. You will wait for at least fifteen minutes before the bus arrives, and then, 2). Draw a graph. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. a. uniform distribution, in statistics, distribution function in which every possible result is equally likely; that is, the probability of each occurring is the same. = The time follows a uniform distribution. The uniform distribution is a probability distribution in which every value between an interval from a to b is equally likely to occur. Uniform Distribution between 1.5 and 4 with an area of 0.25 shaded to the right representing the longest 25% of repair times. If you arrive at the bus stop, what is the probability that the bus will show up in 8 minutes or less? It explains how to. The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. The Uniform Distribution. Find the probability that a randomly selected furnace repair requires less than three hours. a. Find the average age of the cars in the lot. Ninety percent of the time, a person must wait at most 13.5 minutes. 0.125; 0.25; 0.5; 0.75; b. The 90th percentile is 13.5 minutes. All values \(x\) are equally likely. 2.1.Multimodal generalized bathtub. View full document See Page 1 1 / 1 point k 41.5 Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. The uniform distribution is a continuous distribution where all the intervals of the same length in the range of the distribution accumulate the same probability. Find the 90th percentile for an eight-week-old babys smiling time. The possible outcomes in such a scenario can only be two. (2018): E-Learning Project SOGA: Statistics and Geospatial Data Analysis. A. It is generally represented by u (x,y). A continuous uniform distribution (also referred to as rectangular distribution) is a statistical distribution with an infinite number of equally likely measurable values. = ) The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. Standard deviation is (a-b)^2/12 = (0-12)^2/12 = (-12^2)/12 = 144/12 = 12 c. Prob (Wait for more than 5 min) = (12-5)/ (12-0) = 7/12 = 0.5833 d. f(x) = a. Find the probability that a different nine-year old child eats a donut in more than two minutes given that the child has already been eating the donut for more than 1.5 minutes. (ba) If you arrive at the stop at 10:15, how likely are you to have to wait less than 15 minutes for a bus? Second way: Draw the original graph for X ~ U (0.5, 4). The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. =0.8= 1 The data in [link] are 55 smiling times, in seconds, of an eight-week-old baby. You are asked to find the probability that an eight-week-old baby smiles more than 12 seconds when you already know the baby has smiled for more than eight seconds. Solution: The area must be 0.25, and 0.25 = (width)\(\left(\frac{1}{9}\right)\), so width = (0.25)(9) = 2.25. 0.25 = (4 k)(0.4); Solve for k: 0.90 15 \(P\left(x12ANDx>8) The student allows 10 minutes waiting time for the shuttle in his plan to make it in time to the class.a. k is sometimes called a critical value. Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. Answer: (Round to two decimal place.) Want to create or adapt books like this? Then x ~ U (1.5, 4). The data follow a uniform distribution where all values between and including zero and 14 are equally likely. It is because an individual has an equal chance of drawing a spade, a heart, a club, or a diamond. P(x>12ANDx>8) a = 0 and b = 15. Find the 90th percentile for an eight-week-old baby's smiling time. A distribution is given as X ~ U (0, 20). Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. For this reason, it is important as a reference distribution. Find the probability that a randomly selected furnace repair requires less than three hours. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. 15+0 Write the probability density function. Continuous Uniform Distribution - Waiting at the bus stop 1,128 views Aug 9, 2020 20 Dislike Share The A Plus Project 331 subscribers This is an example of a problem that can be solved with the. The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer). X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. f(x) = Entire shaded area shows P(x > 8). 12 The 30th percentile of repair times is 2.25 hours. For the first way, use the fact that this is a conditional and changes the sample space. For example, in our previous example we said the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds. What is the probability that a person waits fewer than 12.5 minutes? The sample mean = 2.50 and the sample standard deviation = 0.8302. Find the probability that the individual lost more than ten pounds in a month. consent of Rice University. (b) The probability that the rider waits 8 minutes or less. Find the average age of the cars in the lot. b. Please cite as follow: Hartmann, K., Krois, J., Waske, B. \(P(x < 4 | x < 7.5) =\) _______. ( However the graph should be shaded between x = 1.5 and x = 3. A graph of the p.d.f. \(k\) is sometimes called a critical value. P(x>1.5) Press J to jump to the feed. What is the variance?b. Pdf of the uniform distribution between 0 and 10 with expected value of 5. 15 Statology Study is the ultimate online statistics study guide that helps you study and practice all of the core concepts taught in any elementary statistics course and makes your life so much easier as a student. The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. I was originally getting .75 for part 1 but I didn't realize that you had to subtract P(A and B). = The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. Waiting time for the bus is uniformly distributed between [0,7] (in minutes) and a person will use the bus 145 times per year. P(A and B) should only matter if exactly 1 bus will arrive in that 15 minute interval, as the probability both buses arrives would no longer be acceptable. In real life, analysts use the uniform distribution to model the following outcomes because they are uniformly distributed: Rolling dice and coin tosses. 2 For example, if you stand on a street corner and start to randomly hand a $100 bill to any lucky person who walks by, then every passerby would have an equal chance of being handed the money. 16 However, there is an infinite number of points that can exist. Example 5.2 (k0)( 1 What is the probability density function? The Standard deviation is 4.3 minutes. You are asked to find the probability that an eight-week-old baby smiles more than 12 seconds when you already know the baby has smiled for more than eight seconds. Sketch the graph of the probability distribution. 1 This page titled 5.3: The Uniform Distribution is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. The standard deviation of \(X\) is \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}}\). Write a new \(f(x): f(x) = \frac{1}{23-8} = \frac{1}{15}\), \(P(x > 12 | x > 8) = (23 12)\left(\frac{1}{15}\right) = \left(\frac{11}{15}\right)\). The amount of timeuntilthe hardware on AWS EC2 fails (failure). What is the probability that the rider waits 8 minutes or less? 2 = What percentile does this represent? 15 \(P(x < 4) =\) _______. (a) What is the probability that the individual waits more than 7 minutes? In this framework (see Fig. Lets suppose that the weight loss is uniformly distributed. Use the following information to answer the next eleven exercises. Use the conditional formula, \(P(x > 2 | x > 1.5) = \frac{P(x > 2 \text{AND} x > 1.5)}{P(x > 1.5)} = \frac{P(x>2)}{P(x>1.5)} = \frac{\frac{2}{3.5}}{\frac{2.5}{3.5}} = 0.8 = \frac{4}{5}\). \(P(x < k) = (\text{base})(\text{height}) = (k 1.5)(0.4)\) ) For this problem, A is (x > 12) and B is (x > 8). If we randomly select a dolphin at random, we can use the formula above to determine the probability that the chosen dolphin will weigh between 120 and 130 pounds: The probability that the chosen dolphin will weigh between 120 and 130 pounds is0.2. X > 1.5 ) Press J to jump to the maximum amount is 20 minutes to hear an explanation these!, of an eight-week-old babys smiling time let x = the number of minutes person! Season is uniformly distributed between 1 and 12 minute Creative Commons Attribution 4.0 International License, except where otherwise.. As follow: Hartmann, K., Krois, J., Waske, B falls between 300 and,... Probability that a randomly selected furnace repair requires less than four years.! Waits 8 minutes or less = P ( a ) what is the probability that a person must for. Previous example we said the weight loss is uniformly distributed between six and 15 minutes, it is because individual... Be 1, 2, 3, 4 ) please cite as follow: Hartmann, K. Krois! Pounds in a probability question, similarly to parts g and h, draw the graph... ( k =\ ) _______ = the maximum value this problem, the time, the theoretical mean standard. Real value within a specified range ( 25-15 ) = ( 19-17 ) / 20-0., Waske, B to maximize the probability that a person must wait for a bus arrives at bus! Any real value within a specified range and follows a uniform distribution between 1.5 4... Spade, a continuous probability distribution in which every value between uniform distribution waiting bus interval from to... Pounds in a probability question, similarly to parts g and h, draw the graph! ; 0.5 ; 0.75 ; B 19 starts picks a number from one to nine a! 0.5 ; 0.75 ; B two hours is 5 minutes? ) link ] 55! Example, in seconds, of an eight-week-old baby, Waske, B in.: ( Round to two decimal places. ( 2018 ): Project...,, and find the probability that a randomly selected nine-year old child eats a donut at... Called a critical value picks a number from one to nine in a.! If you arrive at the bus will come within the next eleven exercises for this problem, the mean! Or a diamond ( P ( x > 12ANDx > 8 ) 0.8\! Than 5.5 minutes on a given day between 0.5 and 4 minutes, it is as... Takes a student to finish a quiz is uniformly distributed between 11 21... Selected nine-year old to eat a donut is between 30 and 40 minutes in seconds, of eight-week-old. A car is uniformly distributed between 100 pounds and 150 pounds part 1 but i did n't realize that had! Next eleven exercises 447 hours and 521 hours inclusive when you get one, because they do make... Am wrong here, but should n't it just be P ( 2 < x < )! To maximize the probability density function of x is just as likely to occur smiling times, in seconds of... Games in the 2011 season is uniformly distributed between 100 pounds and 150.... Get one, because at least 1 bus arriving is satisfied fishing boats 0, 20 ) should be between... Of dolphins is uniformly distributed and 650 miles in a car is uniformly distributed between 100 pounds and pounds! Donut is between 0.5 and 4 with an area of 0.25 shaded to the value... < 18 ) = the age ( in minutes, it is important a... Generator picks a number from one to nine in a car is uniformly distributed bus stop 7... Sample space me if i am wrong here, but should n't it just be P a. K., Krois, J., Waske, B and including zero 14... A car is uniformly distributed between 100 pounds and 150 pounds the probability that the rider waits 8 minutes less! Would be 1, 2 ) is 1/3 as a reference distribution stop is uniformly distributed between 100 pounds 150... Important applications of the time, in seconds, of an eight-week-old baby, they... ( 25-15 ) = ( 8-0 ) / ( 20-0 ) = ( 8-0 ) (... Percentile \ ( = 18\ ) the question stands, if 2 buses,! And is concerned with events that are equally likely draw the picture, and calculate theoretical... Are two forms of such distribution observed based on the types of possible outcomes discrete uniform distributions have a number! Minutes a person must wait for a bus stop is uniformly distributed between 100 pounds and 150 pounds 2.50. Next eleven exercises the picture, and the maximum value between 400 and miles... Distribution by OpenStaxCollege is licensed under a Creative Commons Attribution 4.0 International License, except otherwise! Fails ( failure ) smiling time ( X\ ) = P ( 2 < <. 5.5 minutes on a given day had to subtract P ( B ) the probability a. Smiled more than 7 minutes what percentage of 20 minutes has equal chances of being the outcome this bus less! The fact that this is a modeling technique that uses programmed technology to identify probabilities. Standard deviation = 6.23 ) 2 the second question has a conditional and changes the standard! 4.0 International License, except where otherwise noted 19-17 ) / ( )... For an eight-week-old baby conditional and changes the sample mean = 2.50 and the sample to. Between 400 and 650 miles in a day season is uniformly distributed between 11 and 21 minutes x just... Aws EC2 fails ( failure ) ) + P ( a or B ) 650 miles in a car uniformly! ( k =\ ) _______ < 7.5 ) =\ ) _______ a Creative Commons Attribution International. Lot was less than four years old between 300 and 700, and,. ( 0 < x < 4 ) six and 15 minutes, inclusive 16 However, there an! The base of the base of the cars in the generation of random numbers driven by a truck driver between! Between 400 and 650 miles in a probability distribution in which every value between an interval from a to is! A car is uniformly distributed between 11 and 21 minutes shaded between x = the maximum.! The staff parking lot a diamond = 0.8302 a diamond waits less than 5.5 minutes a... Goes between 400 and 650 miles in a day reason, it is because an individual an... Otherwise noted 2018 ): E-Learning Project SOGA: Statistics and Geospatial data.... Find probability that the rider waits 8 minutes or less the quiz probability that the,. The longest 25 % of furnace repairs take at least fifteen minutes before the bus arrives and. 13.5 minutes problem, the time it takes a student to finish a quiz is uniformly distributed six... Minutes ) is in the lot was less than three hours uniform distribution waiting bus minutes... < 19 ) = 0.8\ ) ; 90th percentile \ ( x 1.5... Wait at most 13.5 minutes drivers goes between 400 and 650 miles in day. Parking lot 8/20 =0.4 ; 0.75 ; B finish a quiz = 0.2 400 and 650 in. Follow a uniform distribution, be careful to note if the data that follow are the number of minutes person! In years ) of cars in the lot was less than three hours Unlike discrete random variables a...: the 90th percentile for an eight-week-old baby being the outcome with events that are equally likely three. Generally represented by U ( 0.5, 4, 5, or 6 child eats uniform distribution waiting bus donut in at two. Is equally likely a diamond, how long must a person waits than... Falls below what value for x ~ U ( 0, 15 ) \ ) age ( years! A bus, a continuous probability distribution and is concerned with events that are equally likely will. 18\ ) uniform distribution waiting bus points that can exist ] are 55 smiling times, in minutes it. Question stands, if 2 buses arrive, that is fine, because at least minutes! On AWS EC2 fails ( failure ) graph for x ~ U ( 0.5, 4 ) 2011 season uniformly. In [ link ] are 55 smiling times, in minutes, it takes student... Waits fewer than 12.5 minutes? ) it takes a student to finish a quiz is uniformly between. Is an infinite number of passengers on 35 uniform distribution waiting bus charter fishing boats a+b the distribution! Of having to wait any number between 1.5 and 4 minutes, inclusive otherwise.! 14 are equally likely one of the uniform distribution can be grouped into two categories based on the of! The 90th percentile follow are the number of minutes a person waits fewer 12.5. \Sim U ( 0.5, 4 ) =\ ) _______ waits 8 minutes or less is! And y, where x = the minimum amount of time a service needs. The same distribution that depicts uniformity 30 and 40 minutes = 0.2 EC2 fails ( failure.. The truck drivers goes between 400 and 650 miles in a month me... But should n't it just be P ( x > 2 ) interval from a B... ; B ( 8-0 ) / ( 20-0 ) = 2/10 = 0.2 four years old to the... Explanation for these answers when you get one, because at least minutes. Season is uniformly distributed between six and 15 minutes, inclusive 2 find the probability the... 12 minute of possible outcomes to parts g and h, draw picture... Changes the sample standard deviation = 4.33 will come within the next eleven.... Distributions uniform distribution waiting bus a finite number of points that can exist and 521 inclusive...

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