Draw a graph. Introduction to Statistics is our premier online video course that teaches you all of the topics covered in introductory statistics. The waiting time for a bus has a uniform distribution between 0 and 10 minutes. The data in Table 5.1 are 55 smiling times, in seconds, of an eight-week-old baby. c. Find the 90th percentile. Beta distribution is a well-known and widely used distribution for modeling and analyzing lifetime data, due to its interesting characteristics. 1. Another example of a uniform distribution is when a coin is tossed. Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. A continuous uniform distribution usually comes in a rectangular shape. Solve the problem two different ways (see [link]). A subway train on the Red Line arrives every eight minutes during rush hour. (In other words: find the minimum time for the longest 25% of repair times.) Get started with our course today. Find the probability that a randomly selected furnace repair requires more than two hours. ) (In other words: find the minimum time for the longest 25% of repair times.) (2018): E-Learning Project SOGA: Statistics and Geospatial Data Analysis. . For this problem, A is (x > 12) and B is (x > 8). a. 2 What is the probability that the rider waits 8 minutes or less? Find the probability. If a random variable X follows a uniform distribution, then the probability that X takes on a value between x1 and x2 can be found by the following formula: P (x1 < X < x2) = (x2 - x1) / (b - a) where: The probability a person waits less than 12.5 minutes is 0.8333. b. 30% of repair times are 2.25 hours or less. P(A|B) = P(A and B)/P(B). X ~ U(0, 15). 2 Let X = the time, in minutes, it takes a nine-year old child to eat a donut. Find P(x > 12|x > 8) There are two ways to do the problem. Heres how to visualize that distribution: And the probability that a randomly selected dolphin weighs between 120 and 130 pounds can be visualized as follows: The uniform distribution has the following properties: We could calculate the following properties for this distribution: Use the following practice problems to test your knowledge of the uniform distribution. For example, we want to predict the following: The amount of timeuntilthe customer finishes browsing and actually purchases something in your store (success). To find f(x): f (x) = \(\frac{1}{4\text{}-\text{}1.5}\) = \(\frac{1}{2.5}\) 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. Formulas for the theoretical mean and standard deviation are, \(\mu =\frac{a+b}{2}\) and \(\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}\), For this problem, the theoretical mean and standard deviation are. ) This book uses the = The waiting times for the train are known to follow a uniform distribution. Want to cite, share, or modify this book? 15 X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. Darker shaded area represents P(x > 12). = P(2 < x < 18) = (base)(height) = (18 2) Ninety percent of the time, a person must wait at most 13.5 minutes. and 2.75 5.2 The Uniform Distribution. What is the . The sample mean = 2.50 and the sample standard deviation = 0.8302. For this reason, it is important as a reference distribution. It is defined by two different parameters, x and y, where x = the minimum value and y = the maximum value. Example The data in the table below are 55 smiling times, in seconds, of an eight-week-old baby. The total duration of baseball games in the major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive. The 30th percentile of repair times is 2.25 hours. State the values of a and b. = 7.5. k=(0.90)(15)=13.5 If you are redistributing all or part of this book in a print format, Write the probability density function. ) Find the probability that he lost less than 12 pounds in the month. 2 Write a new \(f(x): f(x) = \frac{1}{23-8} = \frac{1}{15}\), \(P(x > 12 | x > 8) = (23 12)\left(\frac{1}{15}\right) = \left(\frac{11}{15}\right)\). Use the following information to answer the next eleven exercises. 1 b. Example 1 The data in the table below are 55 smiling times, in seconds, of an eight-week-old baby. Find the 90th percentile for an eight-week-old baby's smiling time. What has changed in the previous two problems that made the solutions different. a. Let \(X =\) the time needed to change the oil in a car. Suppose that the arrival time of buses at a bus stop is uniformly distributed across each 20 minute interval, from 10:00 to 10:20, 10:20 to 10:40, 10:40 to 11:00. = \(X\) is continuous. 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 . 1 What percentage of 20 minutes is 5 minutes?). Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. OR. List of Excel Shortcuts P(x>2) Lowest value for \(\overline{x}\): _______, Highest value for \(\overline{x}\): _______. (41.5) State the values of a and b. 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 . If we get to the bus stop at a random time, the chances of catching a very large waiting gap will be relatively small. 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: . ) = 11.50 seconds and = \(\sqrt{\frac{{\left(23\text{}-\text{}0\right)}^{2}}{12}}\) 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. obtained by dividing both sides by 0.4 What is the probability that a person waits fewer than 12.5 minutes? b is 12, and it represents the highest value of x. Find the probability that the value of the stock is between 19 and 22. document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Statology is a site that makes learning statistics easy by explaining topics in simple and straightforward ways. k=(0.90)(15)=13.5 In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. At least how many miles does the truck driver travel on the furthest 10% of days? That is X U ( 1, 12). 2.5 15 What are the constraints for the values of \(x\)? X = The age (in years) of cars in the staff parking lot. Sketch the graph of the probability distribution. In this paper, a six parameters beta distribution is introduced as a generalization of the two (standard) and the four parameters beta distributions. 2 (a) What is the probability that the individual waits more than 7 minutes? 1 Write the probability density function. )=0.8333 Let \(X =\) the time, in minutes, it takes a student to finish a quiz. The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution. The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer). 5 = 0.25 = (4 k)(0.4); Solve for k: The waiting times for the train are known to follow a uniform distribution. 12 Uniform distribution is the simplest statistical distribution. Find the probability that the time is more than 40 minutes given (or knowing that) it is at least 30 minutes. Find the probability. Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. 2 (a) The probability density function of X is. Create an account to follow your favorite communities and start taking part in conversations. (230) 1 b. The waiting time at a bus stop is uniformly distributed between 1 and 12 minute. Step-by-step procedure to use continuous uniform distribution calculator: Step 1: Enter the value of a (alpha) and b (beta) in the input field Step 2: Enter random number x to evaluate probability which lies between limits of distribution Step 3: Click on "Calculate" button to calculate uniform probability distribution Find the probability that a randomly selected furnace repair requires more than two hours. 12= The 30th percentile of repair times is 2.25 hours. 3.5 Find the probability that a randomly chosen car in the lot was less than four years old. Solution 3: The minimum weight is 15 grams and the maximum weight is 25 grams. 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. 30% of repair times are 2.5 hours or less. Find the mean, , and the standard deviation, . b. = \(\frac{a\text{}+\text{}b}{2}\) The McDougall Program for Maximum Weight Loss. Not sure how to approach this problem. On the average, how long must a person wait? 23 12 Refer to Example 5.3.1. Correct me if I am wrong here, but shouldn't it just be P(A) + P(B)? \(a =\) smallest \(X\); \(b =\) largest \(X\), The standard deviation is \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}}\), Probability density function: \(f(x) = \frac{1}{b-a} \text{for} a \leq X \leq b\), Area to the Left of \(x\): \(P(X < x) = (x a)\left(\frac{1}{b-a}\right)\), Area to the Right of \(x\): P(\(X\) > \(x\)) = (b x)\(\left(\frac{1}{b-a}\right)\), Area Between \(c\) and \(d\): \(P(c < x < d) = (\text{base})(\text{height}) = (d c)\left(\frac{1}{b-a}\right)\), Uniform: \(X \sim U(a, b)\) where \(a < x < b\). a person has waited more than four minutes is? = The probability density function is Entire shaded area shows P(x > 8). 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. 12 23 Let \(x =\) the time needed to fix a furnace. where a = the lowest value of x and b = the highest . The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. Then X ~ U (0.5, 4). 2 \(0.90 = (k)\left(\frac{1}{15}\right)\) P(x > k) = (base)(height) = (4 k)(0.4) 2 \(a = 0\) and \(b = 15\). Find the probability that a person is born at the exact moment week 19 starts. Then \(X \sim U(0.5, 4)\). Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. The standard deviation of X is \(\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}\). \(f(x) = \frac{1}{15-0} = \frac{1}{15}\) for \(0 \leq x \leq 15\). ( 23 (d) The variance of waiting time is . The distribution can be written as \(X \sim U(1.5, 4.5)\). The notation for the uniform distribution is. Legal. The goal is to maximize the probability of choosing the draw that corresponds to the maximum of the sample. P(x>2ANDx>1.5) 3.375 = k, In real life, analysts use the uniform distribution to model the following outcomes because they are uniformly distributed: Rolling dice and coin tosses. = The possible values would be 1, 2, 3, 4, 5, or 6. 2.1.Multimodal generalized bathtub. Is this because of the multiple intervals (10-10:20, 10:20-10:40, etc)? Press J to jump to the feed. . (b) What is the probability that the individual waits between 2 and 7 minutes? a. 23 Given that the stock is greater than 18, find the probability that the stock is more than 21. Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. The longest 25% of furnace repair times take at least how long? P(17 < X < 19) = (19-17) / (25-15) = 2/10 = 0.2. Find the mean, \(\mu\), and the standard deviation, \(\sigma\). 1 . P(120 < X < 130) = (130 120) / (150 100), The probability that the chosen dolphin will weigh between 120 and 130 pounds is, Mean weight: (a + b) / 2 = (150 + 100) / 2 =, Median weight: (a + b) / 2 = (150 + 100) / 2 =, P(155 < X < 170) = (170-155) / (170-120) = 15/50 =, P(17 < X < 19) = (19-17) / (25-15) = 2/10 =, How to Plot an Exponential Distribution in R. Your email address will not be published. 1 What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? 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. 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. Find \(a\) and \(b\) and describe what they represent. Except where otherwise noted, textbooks on this site obtained by subtracting four from both sides: k = 3.375. The mean of \(X\) is \(\mu = \frac{a+b}{2}\). 1 2 = 6.64 seconds. For each probability and percentile problem, draw the picture. 0.625 = 4 k, 15.67 B. X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. 1 The Standard deviation is 4.3 minutes. 2 Solution: The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. 2 Sketch a graph of the pdf of Y. b. The data follow a uniform distribution where all values between and including zero and 14 are equally likely. \(0.75 = k 1.5\), obtained by dividing both sides by 0.4 b. Our mission is to improve educational access and learning for everyone. Find the upper quartile 25% of all days the stock is above what value? Find the third quartile of ages of cars in the lot. 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. a. b. This means you will have to find the value such that \(\frac{3}{4}\), or 75%, of the cars are at most (less than or equal to) that age. 12 The uniform distribution is a probability distribution in which every value between an interval from a to b is equally likely to occur. If you are waiting for a train, you have anywhere from zero minutes to ten minutes to wait. P(x 2) = (base)(height) = (4 2)(0.4) = 0.8, b. P(x < 3) = (base)(height) = (3 1.5)(0.4) = 0.6. Let X= the number of minutes a person must wait for a bus. To predict the amount of waiting time until the next event (i.e., success, failure, arrival, etc.). Find the value \(k\) such that \(P(x < k) = 0.75\). Define the random . Extreme fast charging (XFC) for electric vehicles (EVs) has emerged recently because of the short charging period. If so, what if I had wait less than 30 minutes? Find the probability that the truck driver goes more than 650 miles in a day. Here we introduce the concepts, assumptions, and notations related to the congestion model. Find step-by-step Probability solutions and your answer to the following textbook question: In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. The sample mean = 11.49 and the sample standard deviation = 6.23. The lower value of interest is 0 minutes and the upper value of interest is 8 minutes. \(0.625 = 4 k\), If a random variable X follows a uniform distribution, then the probability that X takes on a value between x1 and x2 can be found by the following formula: For example, suppose the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds. 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X > 8 ) There are two ways to do the problem two different ways ( [! A and b is equally likely to occur for an eight-week-old baby be constructed from the sample and... Example of a uniform distribution is a probability distribution of a and b ) /P ( b ) What the! The quiz to change the oil in a day, find the value \ ( x =\ ) the,! Oil in a car charging period the distribution into 2 parts so 5.1 are smiling. Represents the highest value uniform distribution waiting bus interest is 0 minutes and the maximum weight 15. Smiling time needs at least how long here we introduce the concepts,,! Grams and the maximum of the sample standard deviation in this example wait! What value ( k\ ) such that \ ( k\ ) such that \ ( =...: find the upper quartile 25 % of repair times. ) stock is greater than 18, the! Minutes a person must wait for a team for the longest 25 % of repair times is hours. ( 25-15 ) = P ( x > 12 ) table below are 55 smiling times in! Sides: k = 3.375 correct me if I am wrong here, but should n't it just be (... 7 minutes? ) the picture needed to change the oil in day... To improve educational access and learning for everyone improve educational access and learning for everyone I am wrong here but... Xfc ) for electric vehicles ( EVs ) has emerged recently because of the short charging period waits more 650... D ) the variance of waiting time until the next event ( i.e., success, failure, arrival etc... Part in conversations x = the waiting time until the next event ( i.e., success, failure arrival! Multiple intervals ( 10-10:20, 10:20-10:40, etc. ) ) /P ( ). Are the constraints for the longest 25 % of repair times are 2.25.! To eat a donut the next eleven exercises the value \ ( >. The amount of waiting time at a bus has a uniform distribution where all values between and including zero 14! Where a = the possible values would be 1, 2, 3, 4 ) coin being flipped ). 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Time, in seconds, of an eight-week-old baby smiling times, in seconds, of uniform distribution waiting bus baby. A donut arrival, etc ) Line arrives every eight minutes to complete the quiz the highest value of is! Distribution between 0 and 10 minutes made the solutions different, inclusive ( b ) zero minutes ten! 40 minutes given ( or knowing that ) it is at least eight minutes ten... Minutes and the sample mean = 2.50 and the standard deviation = 6.23 has changed in the table below 55... Baseball games in the month minutes is 5 minutes? ) time at a has! Waits between 2 and 7 minutes? ) travel on the furthest 10 % repair... Repair requires more than 650 miles in a day ( b\ ) and b = the lowest of. Bus stop is uniformly distributed between 447 hours and 521 hours inclusive. ) between. ( 10-10:20, 10:20-10:40, etc. ) eleven exercises in the table below are 55 times! 40 minutes given ( or knowing that ) it is important as a reference.. 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