The probability \(P(c < X < d)\) may be found by computing the area under \(f(x)\), between \(c\) and \(d\). Find the probability that a bus will come within the next 10 minutes. The data that follow are the square footage (in 1,000 feet squared) of 28 homes. a person has waited more than four minutes is? What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? P(2 < x < 18) = (base)(height) = (18 2)\(\left(\frac{1}{23}\right)\) = \(\left(\frac{16}{23}\right)\). a+b In reality, of course, a uniform distribution is . A distribution is given as X ~ U (0, 20). A graph of the p.d.f. \(b\) is \(12\), and it represents the highest value of \(x\). 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). That is X U ( 1, 12). Solution 2: The minimum time is 120 minutes and the maximum time is 170 minutes. Solution 3: The minimum weight is 15 grams and the maximum weight is 25 grams. If we get to the bus stop at a random time, the chances of catching a very large waiting gap will be relatively small. 12 The possible values would be 1, 2, 3, 4, 5, or 6. 11 A student takes the campus shuttle bus to reach the classroom building. A continuous probability distribution is called the uniform distribution and it is related to the events that are equally possible to occur. )( Uniform distribution: happens when each of the values within an interval are equally likely to occur, so each value has the exact same probability as the others over the entire interval givenA Uniform distribution may also be referred to as a Rectangular distribution A continuous uniform distribution is a statistical distribution with an infinite number of equally likely measurable values. Note: Since 25% of repair times are 3.375 hours or longer, that means that 75% of repair times are 3.375 hours or less. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. Given that the stock is greater than 18, find the probability that the stock is more than 21. 11 Find the probability that the individual lost more than ten pounds in a month. Then X ~ U (0.5, 4). The graph of the rectangle showing the entire distribution would remain the same. The probability a person waits less than 12.5 minutes is 0.8333. b. (2018): E-Learning Project SOGA: Statistics and Geospatial Data Analysis. b. Ninety percent of the smiling times fall below the 90th percentile, k, so P(x < k) = 0.90, \(\left(\text{base}\right)\left(\text{height}\right)=0.90\), \(\text{(}k-0\text{)}\left(\frac{1}{23}\right)=0.90\), \(k=\left(23\right)\left(0.90\right)=20.7\). Discrete and continuous are two forms of such distribution observed based on the type of outcome expected. The 90th percentile is 13.5 minutes. Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. Find the probability that she is over 6.5 years old. =45 You already know the baby smiled more than eight seconds. e. It explains how to. Find the probability that a randomly selected furnace repair requires more than two hours. 1.0/ 1.0 Points. It is defined by two different parameters, x and y, where x = the minimum value and y = the maximum value. Find the mean and the standard deviation. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. Solve the problem two different ways (see [link]). X = The age (in years) of cars in the staff parking lot. ) It means every possible outcome for a cause, action, or event has equal chances of occurrence. Find the average age of the cars in the lot. b. P(x>1.5) . Continuous Uniform Distribution Example 2 b. Ninety percent of the smiling times fall below the 90th percentile, \(k\), so \(P(x < k) = 0.90\), \[(k0)\left(\frac{1}{23}\right) = 0.90\]. Let X = the time, in minutes, it takes a student to finish a quiz. What is the probability density function? A good example of a discrete uniform distribution would be the possible outcomes of rolling a 6-sided die. ( 15 e. \(\mu = \frac{a+b}{2}\) and \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}}\), \(\mu = \frac{1.5+4}{2} = 2.75\) hours and \(\sigma = \sqrt{\frac{(4-1.5)^{2}}{12}} = 0.7217\) hours. 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. State the values of a and b. 2 Ace Heating and Air Conditioning Service finds that the amount of time a repairman needs to fix a furnace is uniformly distributed between 1.5 and four hours. Refer to [link]. 15.67 B. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. 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. Write a newf(x): f(x) = \(\frac{1}{23\text{}-\text{8}}\) = \(\frac{1}{15}\), P(x > 12|x > 8) = (23 12)\(\left(\frac{1}{15}\right)\) = \(\left(\frac{11}{15}\right)\). What are the constraints for the values of x? 1. What is the 90th percentile of square footage for homes? The probability P(c < X < d) may be found by computing the area under f(x), between c and d. Since the corresponding area is a rectangle, the area may be found simply by multiplying the width and the height. 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This probability question is a conditional. Then X ~ U (6, 15). This means that any smiling time from zero to and including 23 seconds is equally likely. What does this mean? 2 Write the probability density function. Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. 1 obtained by subtracting four from both sides: k = 3.375. The data in Table \(\PageIndex{1}\) are 55 smiling times, in seconds, of an eight-week-old baby. Find the probability that the time is between 30 and 40 minutes. 15 For the first way, use the fact that this is a conditional and changes the sample space. c. What is the expected waiting time? You are asked to find 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. 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? Let \(X =\) the time, in minutes, it takes a nine-year old child to eat a donut. The amount of timeuntilthe hardware on AWS EC2 fails (failure). X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. The graph illustrates the new sample space. Then \(x \sim U(1.5, 4)\). 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). What is the probability that a person waits fewer than 12.5 minutes? Write the answer in a probability statement. A fireworks show is designed so that the time between fireworks is between one and five seconds, and follows a uniform distribution. = 7.5. The data follow a uniform distribution where all values between and including zero and 14 are equally likely. a+b The Manual on Uniform Traffic Control Devices for Streets and Highways (MUTCD) is incorporated in FHWA regulations and recognized as the national standard for traffic control devices used on all public roads. 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. Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. 1 First way: Since you know the child has already been eating the donut for more than 1.5 minutes, you are no longer starting at a = 0.5 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. If the probability density function or probability distribution of a uniform . Correct me if I am wrong here, but shouldn't it just be P(A) + P(B)? Find the probability that a person is born after week 40. The Standard deviation is 4.3 minutes. 15. Example 5.2 41.5 We write X U(a, b). Let X = the time, in minutes, it takes a nine-year old child to eat a donut. (d) The variance of waiting time is . c. Find the 90th percentile. )( The total duration of baseball games in the major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive. 2 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)\(\left(\frac{1}{b-a}\right)\). As one of the simplest possible distributions, the uniform distribution is sometimes used as the null hypothesis, or initial hypothesis, in hypothesis testing, which is used to ascertain the accuracy of mathematical models. 1 b. 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. and you must attribute OpenStax. P(x>8) P(A or B) = P(A) + P(B) - P(A and B). The 30th percentile of repair times is 2.25 hours. P(x 2|x > 1.5) = admirals club military not in uniform. A uniform distribution has the following properties: The area under the graph of a continuous probability distribution is equal to 1. What is the theoretical standard deviation? f(x) = \(\frac{1}{b-a}\) for a x b. 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. All values x are equally likely. What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds? = a. P(x 2|x > 1.5) = \(\frac{P\left(x>2\text{AND}x>1.5\right)}{P\left(x>\text{1}\text{.5}\right)}=\frac{P\left(x>2\right)}{P\left(x>1.5\right)}=\frac{\frac{2}{3.5}}{\frac{2.5}{3.5}}=\text{0}\text{.8}=\frac{4}{5}\). b. X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. Random sampling because that method depends on population members having equal chances. a. c. This probability question is a conditional. a= 0 and b= 15. = The distribution can be written as X ~ U(1.5, 4.5). 0.90 1 If X has a uniform distribution where a < x < b or a x b, then X takes on values between a and b (may include a and b). Your starting point is 1.5 minutes. You are asked to find 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. Uniform Distribution. Shade the area of interest. This is a conditional probability question. So, mean is (0+12)/2 = 6 minutes b. Another simple example is the probability distribution of a coin being flipped. For this problem, A is (x > 12) and B is (x > 8). (a) The solution is (b) The probability that the rider waits 8 minutes or less. 5 If so, what if I had wait less than 30 minutes? The graph of the rectangle showing the entire distribution would remain the same. 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 \(\frac{4}{5}\). 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. = 11.50 seconds and = \(\sqrt{\frac{{\left(23\text{}-\text{}0\right)}^{2}}{12}}\) A random number generator picks a number from one to nine in a uniform manner. Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. c. Ninety percent of the time, the time a person must wait falls below what value? What is the expected waiting time? Then \(X \sim U(0.5, 4)\). The waiting time for a bus has a uniform distribution between 2 and 11 minutes. This is because of the even spacing between any two arrivals. Formulas for the theoretical mean and standard deviation are, = This is a uniform distribution. Sixty percent of commuters wait more than how long for the train? 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. What percentile does this represent? The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. (b-a)2 Find the 30th percentile for the waiting times (in minutes). Find probability that the time between fireworks is greater than four seconds. . 2 A bus arrives every 10 minutes at a bus stop. Find P(x > 12|x > 8) There are two ways to do the problem. https://openstax.org/books/introductory-statistics/pages/1-introduction, https://openstax.org/books/introductory-statistics/pages/5-2-the-uniform-distribution, Creative Commons Attribution 4.0 International License. We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. Waiting time for the bus is uniformly distributed between [0,7] (in minutes) and a person will use the bus 145 times per year. What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds? Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. ) \(P(x > 2|x > 1.5) = (\text{base})(\text{new height}) = (4 2)(25)\left(\frac{2}{5}\right) =\) ? Solve the problem two different ways (see Example 5.3). The McDougall Program for Maximum Weight Loss. The data that follow are the number of passengers on 35 different charter fishing boats. a+b The data in [link] are 55 smiling times, in seconds, of an eight-week-old baby. For this problem, A is (x > 12) and B is (x > 8). P(x 9). The probability of drawing any card from a deck of cards. c. Find the probability that a random eight-week-old baby smiles more than 12 seconds KNOWING that the baby smiles MORE THAN EIGHT SECONDS. Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. In real life, analysts use the uniform distribution to model the following outcomes because they are uniformly distributed: Rolling dice and coin tosses. 3.375 = k, P(x>2ANDx>1.5) Find the 90th percentile. 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. Uniform distribution has probability density distributed uniformly over its defined interval. 1). Department of Earth Sciences, Freie Universitaet Berlin. 23 What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. 0.90=( P(x > k) = (base)(height) = (4 k)(0.4) Extreme fast charging (XFC) for electric vehicles (EVs) has emerged recently because of the short charging period. Ace Heating and Air Conditioning Service finds that the amount of time a repairman needs to fix a furnace is uniformly distributed between 1.5 and four hours. The probability that a randomly selected nine-year old child eats a donut in at least two minutes is _______. 1.5+4 = Find the probability that the commuter waits less than one minute. The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer). What is the average waiting time (in minutes)? for 1.5 x 4. 0.625 = 4 k, hours and \(\sigma =\sqrt{\frac{{\left(41.5\right)}^{2}}{12}}=0.7217\) hours. Post all of your math-learning resources here. 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. Find the 30th percentile of square footage ( in 1,000 feet squared ) of in. One minute: the area under the graph of the even spacing between any arrivals. 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And probability questions and answers a bus arrives every 10 minutes at a bus has uniform. Between zero and 14 are equally likely maximum time is person waits fewer than minutes. At a bus has a uniform distribution, be careful to note if the data is or... Formula, P ( AANDB ) find the probability that the stock is than. Solution is ( x > 2|x > 1.5 ) find the probability a. Requires more than four seconds a+b the data that follow are the footage... If the probability that a randomly selected student needs uniform distribution waiting bus least eight minutes to complete the quiz x 8! Nine-Year old child eats a donut in at least two minutes is it means every possible outcome for a,. Is greater than four minutes is _______ /2 = 6 minutes b let =! Data in [ link ] ) wait falls below what value correct if...