11 0.90 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. It is _____________ (discrete or continuous). Find the mean, , and the standard deviation, . f(X) = 1 150 = 1 15 for 0 X 15. What is the probability that the rider waits 8 minutes or less? Your email address will not be published. What is the probability that the waiting time for this bus is less than 6 minutes on a given day? When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. It means that the value of x is just as likely to be any number between 1.5 and 4.5. 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. Note: We can use the Uniform Distribution Calculator to check our answers for each of these problems. Solution 1: The minimum amount of time youd have to wait is 0 minutes and the maximum amount is 20 minutes. 5.2 The Uniform Distribution. \(k = (0.90)(15) = 13.5\) Sketch the graph, shade the area of interest. Your probability of having to wait any number of minutes in that interval is the same. 23 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. Uniform Distribution between 1.5 and 4 with an area of 0.30 shaded to the left, representing the shortest 30% of repair times. Let X = the time, in minutes, it takes a student to finish a quiz. What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? 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. a. Solution 3: The minimum weight is 15 grams and the maximum weight is 25 grams. It explains how to. a+b What percentile does this represent? a+b Let k = the 90th percentile. A random number generator picks a number from one to nine in a uniform manner. It would not be described as uniform probability. . It means that the value of x is just as likely to be any number between 1.5 and 4.5. What is P(2 < x < 18)? Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. 2 Press J to jump to the feed. This means that any smiling time from zero to and including 23 seconds is equally likely. For the second way, use the conditional formula from Probability Topics with the original distribution X ~ U (0, 23): P(A|B) = \(\frac{P\left(A\text{AND}B\right)}{P\left(B\right)}\). 15 We write X U(a, b). = \(\frac{0\text{}+\text{}23}{2}\) 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. The mean of X is \(\mu =\frac{a+b}{2}\). What is \(P(2 < x < 18)\)? To find \(f(x): f(x) = \frac{1}{4-1.5} = \frac{1}{2.5}\) so \(f(x) = 0.4\), \(P(x > 2) = (\text{base})(\text{height}) = (4 2)(0.4) = 0.8\), b. For this reason, it is important as a reference distribution. (b-a)2 . 41.5 obtained by dividing both sides by 0.4 P(x12ANDx>8) a person has waited more than four minutes is? Find the 90th percentile. 2 15 The 90th percentile is 13.5 minutes. Find the 90th percentile for an eight-week-old baby's smiling time. Thank you! 1. A continuous uniform distribution usually comes in a rectangular shape. If you randomly select a frog, what is the probability that the frog weighs between 17 and 19 grams? Write the probability density function. What has changed in the previous two problems that made the solutions different? 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). are not subject to the Creative Commons license and may not be reproduced without the prior and express written The waiting time for a bus has a uniform distribution between 0 and 10 minutes. Commuting to work requiring getting on a bus near home and then transferring to a second bus. Find the probability that the time is more than 40 minutes given (or knowing that) it is at least 30 minutes. Write a new \(f(x): f(x) = \frac{1}{23-8} = \frac{1}{15}\), \(P(x > 12 | x > 8) = (23 12)\left(\frac{1}{15}\right) = \left(\frac{11}{15}\right)\). 233K views 3 years ago This statistics video provides a basic introduction into continuous probability distribution with a focus on solving uniform distribution problems. 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. (a) The probability density function of X is. \(P\left(x 9). consent of Rice University. 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. The data that follow are the square footage (in 1,000 feet squared) of 28 homes. = 11.50 seconds and = \(\sqrt{\frac{{\left(23\text{}-\text{}0\right)}^{2}}{12}}\) b is 12, and it represents the highest value of x. Find \(P(x > 12 | x > 8)\) There are two ways to do the problem. The probability a person waits less than 12.5 minutes is 0.8333. b. = Find the probability that the value of the stock is more than 19. 3.375 hours is the 75th percentile of furnace repair times. \(0.625 = 4 k\), a. It means every possible outcome for a cause, action, or event has equal chances of occurrence. =0.8= Then \(x \sim U(1.5, 4)\). Let \(X =\) the time needed to change the oil on a car. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. c. Ninety percent of the time, the time a person must wait falls below what value? Let \(X =\) the number of minutes a person must wait for a bus. Figure Press question mark to learn the rest of the keyboard shortcuts. a= 0 and b= 15. Correct me if I am wrong here, but shouldn't it just be P(A) + P(B)? = 1.5+4 Write the probability density function. Then X ~ U (6, 15). For example, we want to predict the following: The amount of timeuntilthe customer finishes browsing and actually purchases something in your store (success). 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 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. 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. 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). = ) ) a. Learn more about how Pressbooks supports open publishing practices. You already know the baby smiled more than eight seconds. Uniform distribution is the simplest statistical distribution. 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. The Uniform Distribution by OpenStaxCollege is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted. 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. Sketch the graph of the probability distribution. 0.125; 0.25; 0.5; 0.75; b. Question 2: The length of an NBA game is uniformly distributed between 120 and 170 minutes. The probability density function of X is \(f\left(x\right)=\frac{1}{b-a}\) for a x b. Sixty percent of commuters wait more than how long for the train? It can provide a probability distribution that can guide the business on how to properly allocate the inventory for the best use of square footage. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. 0.625 = 4 k, P(x>8) (ba) What is the probability density function? Example 5.2 a+b b. The second question has a conditional probability. Your starting point is 1.5 minutes. P(A|B) = P(A and B)/P(B). (a) What is the probability that the individual waits more than 7 minutes? d. What is standard deviation of waiting time? = 2.5 The waiting time for a bus has a uniform distribution between 0 and 8 minutes. The notation for the uniform distribution is. The waiting times for the train are known to follow a uniform distribution. 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. They can be said to follow a uniform distribution from one to 53 (spread of 52 weeks). pdf: \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\), standard deviation \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}}\), \(P(c < X < d) = (d c)\left(\frac{1}{b-a}\right)\). The Bus wait times are uniformly distributed between 5 minutes and 23 minutes. 1 = ) c. What is the expected waiting time? Uniform Distribution between 1.5 and 4 with an area of 0.30 shaded to the left, representing the shortest 30% of repair times. What are the constraints for the values of x? ba P(x>2) We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. List of Excel Shortcuts a form of probability distribution where all values between and including seconds! Of games for a train to arrive the solutions different if the data that follow are the square footage in... 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