Let X = the time, in minutes, it takes a student to finish a quiz. Possible waiting times are along the horizontal axis, and the vertical axis represents the probability. hours and In statistics and probability theory, a discrete uniform distribution is a statistical distribution where the probability of outcomes is equally likely and with finite values. Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. P(x>12) Solution 2: The minimum time is 120 minutes and the maximum time is 170 minutes. for 0 X 23. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. The McDougall Program for Maximum Weight Loss. 2 P(x2ANDx>1.5) f(x) = ( Find the probability that she is between four and six years old. Sketch the graph, shade the area of interest. Solution 1: The minimum amount of time youd have to wait is 0 minutes and the maximum amount is 20 minutes. 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. Your starting point is 1.5 minutes. Let \(X =\) the time needed to change the oil in a car. The total duration of baseball games in the major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Since the corresponding area is a rectangle, the area may be found simply by multiplying the width and the height. It means every possible outcome for a cause, action, or event has equal chances of occurrence. 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: Develop analytical superpowers by learning how to use programming and data analytics tools such as VBA, Python, Tableau, Power BI, Power Query, and more. X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. That is . 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. = 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. Except where otherwise noted, textbooks on this site If you are redistributing all or part of this book in a print format, A student takes the campus shuttle bus to reach the classroom building. 2 The graph of the rectangle showing the entire distribution would remain the same. 15 Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. 2 So, P(x > 21|x > 18) = (25 21)\(\left(\frac{1}{7}\right)\) = 4/7. 2 23 Let x = the time needed to fix a furnace. In statistics, uniform distribution is a probability distribution where all outcomes are equally likely. For the second way, use the conditional formula from Probability Topics with the original distribution X ~ U (0, 23): P(A|B) = 5.2 The Uniform Distribution. \(P(x > k) = 0.25\) 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. Draw a graph. P(x>1.5) OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. P(x > k) = 0.25 The age of cars in the staff parking lot of a suburban college is uniformly distributed from six months (0.5 years) to 9.5 years. 1. 1 Can you take it from here? 4 P(2 < x < 18) = (base)(height) = (18 2) The amount of timeuntilthe hardware on AWS EC2 fails (failure). Legal. = If we get to the bus stop at a random time, the chances of catching a very large waiting gap will be relatively small. 2 Write a new f(x): f(x) = Find the probability that the value of the stock is between 19 and 22. The sample mean = 7.9 and the sample standard deviation = 4.33. Find the mean, \(\mu\), and the standard deviation, \(\sigma\). What does this mean? Uniform Distribution between 1.5 and 4 with an area of 0.25 shaded to the right representing the longest 25% of repair times. Sketch the graph of the probability distribution. 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. 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. 23 The sample mean = 2.50 and the sample standard deviation = 0.8302. Let k = the 90th percentile. 1 Correct answers: 3 question: The waiting time for a bus has a uniform distribution between 0 and 8 minutes. 30% of repair times are 2.5 hours or less. In their calculations of the optimal strategy . 15 A subway train on the Red Line arrives every eight minutes during rush hour. It is defined by two different parameters, x and y, where x = the minimum value and y = the maximum value. P(x>1.5) 11 12 = What is the probability that a person waits fewer than 12.5 minutes? 2 You can do this two ways: Draw the graph where a is now 18 and b is still 25. Another example of a uniform distribution is when a coin is tossed. P(x > 2|x > 1.5) = (base)(new height) = (4 2)\(\left(\frac{2}{5}\right)\)= ? 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 . b is 12, and it represents the highest value of x. )=0.8333 State the values of a and b. P(B) Shade the area of interest. Find P(x > 12|x > 8) There are two ways to do the problem. 0.90 The mean of \(X\) is \(\mu = \frac{a+b}{2}\). The mean of X is \(\mu =\frac{a+b}{2}\). It is generally denoted by u (x, y). uniform distribution, in statistics, distribution function in which every possible result is equally likely; that is, the probability of each occurring is the same. a. k is sometimes called a critical value. 150 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 (also referred to as rectangular distribution) is a statistical distribution with an infinite number of equally likely measurable values. What is the probability density function? = \(\frac{6}{9}\) = \(\frac{2}{3}\). percentile of this distribution? Discrete and continuous are two forms of such distribution observed based on the type of outcome expected. 5 Correct me if I am wrong here, but shouldn't it just be P(A) + P(B)? Commuting to work requiring getting on a bus near home and then transferring to a second bus. The sample mean = 2.50 and the sample standard deviation = 0.8302. The data in [link] are 55 smiling times, in seconds, of an eight-week-old baby. Uniform Distribution between 1.5 and 4 with an area of 0.30 shaded to the left, representing the shortest 30% of repair times. a. Department of Earth Sciences, Freie Universitaet Berlin. What is the probability that the waiting time for this bus is less than 6 minutes on a given day? e. b. Ninety percent of the smiling times fall below the 90th percentile, \(k\), so \(P(x < k) = 0.90\), \[(k0)\left(\frac{1}{23}\right) = 0.90\]. In words, define the random variable \(X\). for 0 x 15. 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. What is the 90th . Use the following information to answer the next eight exercises. \(k = 2.25\) , obtained by adding 1.5 to both sides. The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer). 1 0.625 = 4 k, 2 The time (in minutes) until the next bus departs a major bus depot follows a distribution with f(x) = 1 20. where x goes from 25 to 45 minutes. = P(A|B) = P(A and B)/P(B). Then \(X \sim U(0.5, 4)\). = \(\sqrt{\frac{\left(b-a{\right)}^{2}}{12}}=\sqrt{\frac{\left(\mathrm{15}-0{\right)}^{2}}{12}}\) = 4.3. \(f\left(x\right)=\frac{1}{8}\) where \(1\le x\le 9\). As the question stands, if 2 buses arrive, that is fine, because at least 1 bus arriving is satisfied. The sample mean = 7.9 and the sample standard deviation = 4.33. The second question has a conditional probability. 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 . View full document See Page 1 1 / 1 point (d) The variance of waiting time is . You must reduce the sample space. 0+23 b. Unlike discrete random variables, a continuous random variable can take any real value within a specified range. P(B). Find the mean and the standard deviation. The Bus wait times are uniformly distributed between 5 minutes and 23 minutes. and (b) The probability that the rider waits 8 minutes or less. We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. = P(x>12ANDx>8) where a = the lowest value of x and b = the highest . P(155 < X < 170) = (170-155) / (170-120) = 15/50 = 0.3. b. You must reduce the sample space. d. What is standard deviation of waiting time? \(P(x > 2|x > 1.5) = (\text{base})(\text{new height}) = (4 2)(25)\left(\frac{2}{5}\right) =\) ? a+b = On the average, a person must wait 7.5 minutes. a = 0 and b = 15. Example 5.2 Use the conditional formula, P(x > 2|x > 1.5) = The probability of drawing any card from a deck of cards. Find the probability that a person is born at the exact moment week 19 starts. 5 . \(X =\) __________________. Let k = the 90th percentile. Use the conditional formula, \(P(x > 2 | x > 1.5) = \frac{P(x > 2 \text{AND} x > 1.5)}{P(x > 1.5)} = \frac{P(x>2)}{P(x>1.5)} = \frac{\frac{2}{3.5}}{\frac{2.5}{3.5}} = 0.8 = \frac{4}{5}\). Write a newf(x): f(x) = \(\frac{1}{23\text{}-\text{8}}\) = \(\frac{1}{15}\), P(x > 12|x > 8) = (23 12)\(\left(\frac{1}{15}\right)\) = \(\left(\frac{11}{15}\right)\). Let \(k =\) the 90th percentile. Let X = the time needed to change the oil on a car. Creative Commons Attribution License 15 Uniform distribution refers to the type of distribution that depicts uniformity. \(P(x < 4 | x < 7.5) =\) _______. 12= Want to cite, share, or modify this book? are not subject to the Creative Commons license and may not be reproduced without the prior and express written One of the most important applications of the uniform distribution is in the generation of random numbers. 41.5 \(b\) is \(12\), and it represents the highest value of \(x\). 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. Write the probability density function. 5 Then x ~ U (1.5, 4). Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. 23 = A distribution is given as X ~ U (0, 20). What percentile does this represent? The notation for the uniform distribution is. citation tool such as. 23 1 Solution 3: The minimum weight is 15 grams and the maximum weight is 25 grams. 1 233K views 3 years ago This statistics video provides a basic introduction into continuous probability distribution with a focus on solving uniform distribution problems. 1 The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution. 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. The probability a person waits less than 12.5 minutes is 0.8333. b. =0.7217 The data that follow are the square footage (in 1,000 feet squared) of 28 homes. What is the probability that a bus will come in the first 10 minutes given that it comes in the last 15 minutes (i.e. Find the probability that a randomly selected furnace repair requires more than two hours. 1 What is the 90th percentile of square footage for homes? Find probability that the time between fireworks is greater than four seconds. This is a uniform distribution. The student allows 10 minutes waiting time for the shuttle in his plan to make it in time to the class.a. 23 Draw a graph. Random sampling because that method depends on population members having equal chances. Formulas for the theoretical mean and standard deviation are, = Shade the area of interest. admirals club military not in uniform. = The Continuous Uniform Distribution in R. You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License. f(x) = \(\frac{1}{b-a}\) for a x b. Let X = the time, in minutes, it takes a nine-year old child to eat a donut. What is the height of f(x) for the continuous probability distribution? Solution: ) P(x
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