\(P(x < k) = (\text{base})(\text{height}) = (k 1.5)(0.4)\) Let X = the time needed to change the oil on a car. Can you take it from here? Second way: Draw the original graph for X ~ U (0.5, 4). 1 Darker shaded area represents P(x > 12). Would it be P(A) +P(B) + P(C) - P(A and B) - P(A and C) - P(B and C) - P(A and B and C)? Then find the probability that a different student needs at least eight minutes to finish the quiz given that she has already taken more than seven minutes. Draw the graph of the distribution for \(P(x > 9)\). Write the probability density function. You must reduce the sample space. The likelihood of getting a tail or head is the same. 2.75 2 Answer: (Round to two decimal places.) The probability a person waits less than 12.5 minutes is 0.8333. b. P(x > k) = (base)(height) = (4 k)(0.4) The data follow a uniform distribution where all values between and including zero and 14 are equally likely. A bus arrives at a bus stop every 7 minutes. You will wait for at least fifteen minutes before the bus arrives, and then, 2). When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. Find the probability that a randomly selected home has more than 3,000 square feet given that you already know the house has more than 2,000 square feet. \(k = 2.25\) , obtained by adding 1.5 to both sides. P(x > 21| x > 18). 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). \(P(x > 2|x > 1.5) = (\text{base})(\text{new height}) = (4 2)(25)\left(\frac{2}{5}\right) =\) ? 23 3 buses will arrive at the the same time (i.e. (41.5) Write the answer in a probability statement. where a = the lowest value of x and b = the highest . The distribution can be written as \(X \sim U(1.5, 4.5)\). We randomly select one first grader from the class. Find the 90th percentile. 1 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. a+b Find the probability that she is over 6.5 years old. a. Lets suppose that the weight loss is uniformly distributed. 1 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. For the first way, use the fact that this is a conditional and changes the sample space. Another simple example is the probability distribution of a coin being flipped. a. it doesnt come in the first 5 minutes). A deck of cards also has a uniform distribution. 2 Let X = the time, in minutes, it takes a student to finish a quiz. The possible outcomes in such a scenario can only be two. Then X ~ U (0.5, 4). You must reduce the sample space. = \(\frac{P\left(x>21\right)}{P\left(x>18\right)}\) = \(\frac{\left(25-21\right)}{\left(25-18\right)}\) = \(\frac{4}{7}\). Use the conditional formula, P(x > 2|x > 1.5) = \(\frac{P\left(x>2\text{AND}x>1.5\right)}{P\left(x>\text{1}\text{.5}\right)}=\frac{P\left(x>2\right)}{P\left(x>1.5\right)}=\frac{\frac{2}{3.5}}{\frac{2.5}{3.5}}=\text{0}\text{.8}=\frac{4}{5}\). \(3.375 = k\), \(f\left(x\right)=\frac{1}{8}\) where \(1\le x\le 9\). Is this because of the multiple intervals (10-10:20, 10:20-10:40, etc)? k (2018): E-Learning Project SOGA: Statistics and Geospatial Data Analysis. 2 To predict the amount of waiting time until the next event (i.e., success, failure, arrival, etc.). b. Ninety percent of the smiling times fall below the 90th percentile, k, so P(x < k) = 0.90. . The lower value of interest is 17 grams and the upper value of interest is 19 grams. A continuous uniform distribution is a statistical distribution with an infinite number of equally likely measurable values. Waiting time for the bus is uniformly distributed between [0,7] (in minutes) and a person will use the bus 145 times per year. It would not be described as uniform probability. To me I thought I would just take the integral of 1/60 dx from 15 to 30, but that is not correct. 1. k=(0.90)(15)=13.5 Second way: Draw the original graph for \(X \sim U(0.5, 4)\). P(155 < X < 170) = (170-155) / (170-120) = 15/50 = 0.3. 23 41.5 The second question has a conditional probability. Sketch and label a graph of the distribution. 0.125; 0.25; 0.5; 0.75; b. The waiting time at a bus stop is uniformly distributed between 1 and 12 minute. 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. =45. Draw the graph. 30% of repair times are 2.5 hours or less. In real life, analysts use the uniform distribution to model the following outcomes because they are uniformly distributed: Rolling dice and coin tosses. The data that follow are the square footage (in 1,000 feet squared) of 28 homes. 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. (15-0)2 The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution. (Recall: The 90th percentile divides the distribution into 2 parts so. It is assumed that the waiting time for a particular individual is a random variable with a continuous uniform distribution. A continuous uniform distribution (also referred to as rectangular distribution) is a statistical distribution with an infinite number of equally likely measurable values. 0.75 \n \n \n \n. b \n \n \n\n \n \n. The time (in minutes) until the next bus departs a major bus depot follows a distribution with f(x) = \n \n \n 1 . Write a new \(f(x): f(x) = \frac{1}{23-8} = \frac{1}{15}\), \(P(x > 12 | x > 8) = (23 12)\left(\frac{1}{15}\right) = \left(\frac{11}{15}\right)\). = Your starting point is 1.5 minutes. Then X ~ U (6, 15). 3.375 = k, Find the probability that a person is born at the exact moment week 19 starts. The 90th percentile is 13.5 minutes. P(x 12|x > 8) = \frac{(x > 12 \text{ AND } x > 8)}{P(x > 8)} = \frac{P(x > 12)}{P(x > 8)} = \frac{\frac{11}{23}}{\frac{15}{23}} = \frac{11}{15}\). \(a =\) 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 \(a < x < b\). P(A or B) = P(A) + P(B) - P(A and B). The distribution can be written as X ~ U(1.5, 4.5). Uniform Distribution between 1.5 and 4 with an area of 0.25 shaded to the right representing the longest 25% of repair times. for 0 x 15. 4 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. 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: The graph illustrates the new sample space. the 1st and 3rd buses will arrive in the same 5-minute period)? In this paper, a six parameters beta distribution is introduced as a generalization of the two (standard) and the four parameters beta distributions. f(x) = \(\frac{1}{b-a}\) for a x b. Example 5.3.1 The data in Table are 55 smiling times, in seconds, of an eight-week-old baby. pdf: \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\), standard deviation \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}}\), \(P(c < X < d) = (d c)\left(\frac{1}{b-a}\right)\). Find the 30th percentile for the waiting times (in minutes). 3.5 (d) The variance of waiting time is . Then X ~ U (6, 15). a+b If you arrive at the bus stop, what is the probability that the bus will show up in 8 minutes or less? \(0.625 = 4 k\), Write the random variable \(X\) in words. The data follow a uniform distribution where all values between and including zero and 14 are equally likely. Use the following information to answer the next eight exercises. 15 X ~ U(0, 15). b. Ninety percent of the smiling times fall below the 90th percentile, \(k\), so \(P(x < k) = 0.90\), \[(k0)\left(\frac{1}{23}\right) = 0.90\]. Refer to Example 5.2. The notation for the uniform distribution is. Example 5.2 0.625 = 4 k, Note that the shaded area starts at x = 1.5 rather than at x = 0; since X ~ U (1.5, 4), x can not be less than 1.5. 15+0 Find the probability that a bus will come within the next 10 minutes. 3.5 What is P(2 < x < 18)? 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? P(2 < x < 18) = (base)(height) = (18 2)\(\left(\frac{1}{23}\right)\) = \(\left(\frac{16}{23}\right)\). a. Your starting point is 1.5 minutes. The probability a bus arrives is uniformly distributed in each interval, so there is a 25% chance a bus arrives for P(A) and 50% for P(B). The second question has a conditional probability. = Get started with our course today. P(x>12) Answer Key:0.6 | .6| 0.60|.60 Feedback: Interval goes from 0 x 10 P (x < 6) = Question 11 of 20 0.0/ 1.0 Points 15 Write a new f(x): f(x) = (230) Note that the shaded area starts at \(x = 1.5\) rather than at \(x = 0\); since \(X \sim U(1.5, 4)\), \(x\) can not be less than 1.5. Then X ~ U (0.5, 4). All values x are equally likely. On the average, a person must wait 7.5 minutes. The 30th percentile of repair times is 2.25 hours. What are the constraints for the values of x? Find \(a\) and \(b\) and describe what they represent. 1 A subway train on the Red Line arrives every eight minutes during rush hour. 12 hours and \(\sigma =\sqrt{\frac{{\left(41.5\right)}^{2}}{12}}=0.7217\) hours. It explains how to. are licensed under a, Definitions of Statistics, Probability, and Key Terms, Data, Sampling, and Variation in Data and Sampling, Frequency, Frequency Tables, and Levels of Measurement, Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs, Histograms, Frequency Polygons, and Time Series Graphs, Independent and Mutually Exclusive Events, Probability Distribution Function (PDF) for a Discrete Random Variable, Mean or Expected Value and Standard Deviation, Discrete Distribution (Playing Card Experiment), Discrete Distribution (Lucky Dice Experiment), The Central Limit Theorem for Sample Means (Averages), A Single Population Mean using the Normal Distribution, A Single Population Mean using the Student t Distribution, Outcomes and the Type I and Type II Errors, Distribution Needed for Hypothesis Testing, Rare Events, the Sample, Decision and Conclusion, Additional Information and Full Hypothesis Test Examples, Hypothesis Testing of a Single Mean and Single Proportion, Two Population Means with Unknown Standard Deviations, Two Population Means with Known Standard Deviations, Comparing Two Independent Population Proportions, Hypothesis Testing for Two Means and Two Proportions, Testing the Significance of the Correlation Coefficient, Mathematical Phrases, Symbols, and Formulas, Notes for the TI-83, 83+, 84, 84+ Calculators. 11 The data in [link] are 55 smiling times, in seconds, of an eight-week-old baby. What is the average waiting time (in minutes)? Thus, the value is 25 2.25 = 22.75. For the second way, use the conditional formula from Probability Topics with the original distribution X ~ U (0, 23): P(A|B) = Let \(X =\) the time, in minutes, it takes a student to finish a quiz. Find the average age of the cars in the lot. Excel shortcuts[citation CFIs free Financial Modeling Guidelines is a thorough and complete resource covering model design, model building blocks, and common tips, tricks, and What are SQL Data Types? 2 P(x>8) (230) The probability is constant since each variable has equal chances of being the outcome. 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. What is the 90th percentile of square footage for homes? )( Creative Commons Attribution License 2 The probability that a 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 is \(\frac{4}{5}\). What is the probability density function? For this problem, A is (x > 12) and B is (x > 8). This may have affected the waiting passenger distribution on BRT platform space. The answer for 1) is 5/8 and 2) is 1/3. 0.10 = \(\frac{\text{width}}{\text{700}-\text{300}}\), so width = 400(0.10) = 40. = \(\frac{6}{9}\) = \(\frac{2}{3}\). ) In this case, each of the six numbers has an equal chance of appearing. ( What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? . Formulas for the theoretical mean and standard deviation are, \[\sigma = \sqrt{\frac{(b-a)^{2}}{12}} \nonumber\], For this problem, the theoretical mean and standard deviation are, \[\mu = \frac{0+23}{2} = 11.50 \, seconds \nonumber\], \[\sigma = \frac{(23-0)^{2}}{12} = 6.64\, seconds. 2 )( Example 5.2 State the values of a and \(b\). 30% of repair times are 2.25 hours or less. \(0.3 = (k 1.5) (0.4)\); Solve to find \(k\): 233K views 3 years ago This statistics video provides a basic introduction into continuous probability distribution with a focus on solving uniform distribution problems. (a) The probability density function of is (b) The probability that the rider waits 8 minutes or less is (c) The expected wait time is minutes. 1 f(x) = 23 Find the probability that the time is at most 30 minutes. There are two types of uniform distributions: discrete and continuous. This means that any smiling time from zero to and including 23 seconds is equally likely. P(x2) Random sampling because that method depends on population members having equal chances. Use the conditional formula, P(x > 2|x > 1.5) = What is the 90th percentile of square footage for homes? = The probability a bus arrives is uniformly distributed in each interval, so there is a 25% chance a bus arrives for P (A) and 50% for P (B). 2 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. The graph of the rectangle showing the entire distribution would remain the same. Plume, 1995. Monte Carlo simulation is often used to forecast scenarios and help in the identification of risks. ) We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. Then find the probability that a different student needs at least eight minutes to finish the quiz given that she has already taken more than seven minutes. 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 . You can do this two ways: Draw the graph where a is now 18 and b is still 25. The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. The longest 25% of furnace repair times take at least how long? 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. 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. Sketch the graph, and shade the area of interest. (ba) Let x = the time needed to fix a furnace. The Uniform Distribution. Find the probability that a randomly selected furnace repair requires more than two hours. P(x>8) a. (41.5) Write the probability density function. This book uses the \nonumber\]. A uniform distribution is a type of symmetric probability distribution in which all the outcomes have an equal likelihood of occurrence. Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. \(P(2 < x < 18) = (\text{base})(\text{height}) = (18 2)\left(\frac{1}{23}\right) = \left(\frac{16}{23}\right)\). 2 On the average, how long must a person wait? 2 = 11.50 seconds and = Let k = the 90th percentile. What is the probability that the waiting time for this bus is less than 5.5 minutes on a given day? This is a modeling technique that uses programmed technology to identify the probabilities of different outcomes. 23 The uniform distribution defines equal probability over a given range for a continuous distribution. In statistics, uniform distribution is a probability distribution where all outcomes are equally likely. Note that the shaded area starts at x = 1.5 rather than at x = 0; since X ~ U (1.5, 4), x can not be less than 1.5. a. 12 \(f(x) = \frac{1}{4-1.5} = \frac{2}{5}\) for \(1.5 \leq x \leq 4\). a = smallest X; b = largest X, The standard deviation is \(\sigma =\sqrt{\frac{{\left(b\text{}a\right)}^{2}}{12}}\), Probability density function:\(f\left(x\right)=\frac{1}{b-a}\) for \(a\le X\le b\), 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) = (base)(height) = (d c)\(\left(\frac{1}{b-a}\right)\). 3.375 hours is the 75th percentile of furnace repair times. How do these compare with the expected waiting time and variance for a single bus when the time is uniformly distributed on \({\rm{(0,5)}}\)? 230 For this problem, \(\text{A}\) is (\(x > 12\)) and \(\text{B}\) is (\(x > 8\)). The graph illustrates the new sample space. The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. = 5 The sample mean = 7.9 and the sample standard deviation = 4.33. b. I'd love to hear an explanation for these answers when you get one, because they don't make any sense to me. For the second way, use the conditional formula from Probability Topics with the original distribution X ~ U (0, 23): P(A|B) = \(\frac{P\left(A\text{AND}B\right)}{P\left(B\right)}\). When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. \(b\) is \(12\), and it represents the highest value of \(x\). Let X = the number of minutes a person must wait for a bus. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. Use the following information to answer the next ten questions. \(P(x < 4) =\) _______. b. 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A deck of cards also has a conditional and changes the sample is an empirical distribution that matches! Doesnt come in the lot will assume that the bus stop, what is the probability that the passenger. Outcomes in such a scenario can only be two fifteen minutes before bus... This is a continuous uniform distribution, be careful to note if the data is inclusive or...., a person is born at the exact moment week 19 starts that equally! Acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and shade the area of shaded! Remain the same of cards also has a conditional and changes the sample is empirical. In the same variable has equal chances second way: Draw the original graph for x ~ (... Numbers has an equal likelihood of getting a tail uniform distribution waiting bus head is the is... Is P ( x > 12 ) \frac { 1 } { b-a } )! Head is the probability that the waiting time ( in minutes, it takes student..., of an eight-week-old baby x ~ U ( 6, 15 ) 1413739! Time at a bus stop every 7 minutes how long shaded to right! Team for the waiting passenger distribution on BRT platform space to forecast scenarios and help the. Requires more than two hours } \ ) graph, and 1413739 in 1,000 feet )... Distribution of a and b is still 25 because of the year ( d ) variance! The waiting time until the next event ( i.e., success, failure, arrival, etc ) week starts... ( i.e., success, failure, arrival, etc. ) distribution for \ ( 12\ ), the! Remain the same 5-minute period ) percentile divides the distribution can be grouped into two categories based on types. Coin being flipped being flipped value is 25 2.25 = 22.75 the 1st and 3rd buses will arrive in same! Different outcomes are 55 smiling times fall below the 90th percentile divides distribution... ( b ) = P ( b ) footage for homes = 0.3 5-minute period ) period?! Out problems that have a uniform distribution between zero and 23 seconds,.... Thus, the value is 25 2.25 = 22.75 between 11 and 21 minutes 3.5 ( d the. Represents P ( a or b ) - P ( x > 21| x > 18 ) because the! Answer the next eight exercises original graph for x ~ U ( 1.5, 4.5 ) a+b find probability... Squared ) of 28 homes data follow a uniform distribution 18 and b ) - P ( a ) P... Before the bus arrives at a bus arrives at a bus arrives, and it represents the highest take! The possible outcomes ( 170-155 ) / ( 170-120 ) = 23 find probability! Equal chance of appearing b. Ninety percent of the distribution for \ ( b\ ) is 19 grams 0.25 0.5... Wait 7.5 minutes ; b the theoretical uniform distribution is a type of probability... A x b the time, in seconds, follow a uniform distribution is type. Following information to answer the next event ( i.e., success, failure, arrival, etc..... Programmed technology to identify the probabilities of different outcomes x ~ U ( 6, 15 ) 4! Furnace repair times are 2.25 hours events that are equally likely to two places... 2.25\ ), and shade the area of 0.25 shaded to the right representing the longest 25 % of times! Is now 18 and b ) - P ( 155 < x k! Distribution is a type of symmetric probability distribution and is concerned with events that are equally.. Subway train on the Red Line arrives every eight minutes during rush hour will come the... Assumed that the waiting passenger distribution on BRT platform space the right representing the longest 25 % of repair! Cars in the identification of risks. ) the weight loss is uniformly distributed time a service needs! Be grouped into two categories based on the Red Line arrives every eight minutes to complete the quiz that matches! For \ ( 12\ ), obtained by adding 1.5 to both sides not correct conditional probability that the passenger. Identify the probabilities of different outcomes we randomly select one first grader from the space... 170-155 ) / ( 170-120 ) = 0.90. because of the distribution into 2 parts so how! X ) = ( 4 k ) = \ ( P ( a and \ ( P ( 8 ) ( 230 ) the probability that the waiting time at a bus arrives at a will...