The Standard deviation is 4.3 minutes. =0.7217 2.1.Multimodal generalized bathtub. (d) The variance of waiting time is . Second way: Draw the original graph for X ~ U (0.5, 4). Let \(X =\) the time needed to change the oil in a car. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. hours. 41.5 1.5+4 2 The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. Find the third quartile of ages of cars in the lot. \(0.25 = (4 k)(0.4)\); Solve for \(k\): b. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo \(P(x < 4 | x < 7.5) =\) _______. The 90th percentile is 13.5 minutes. For example, it can arise in inventory management in the study of the frequency of inventory sales. 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). To predict the amount of waiting time until the next event (i.e., success, failure, arrival, etc.). The waiting time at a bus stop is uniformly distributed between 1 and 12 minute. pdf: \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\), standard deviation \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}}\), \(P(c < X < d) = (d c)\left(\frac{1}{b-a}\right)\). Post all of your math-learning resources here. P(x>2ANDx>1.5) 5. c. Find the 90th percentile. 1 12 = Find the probability that a randomly selected furnace repair requires more than two hours. The probability density function is 15 Uniform distribution is the simplest statistical distribution. Solution: 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 Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. = When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. You already know the baby smiled more than eight seconds. Find the probability that the commuter waits between three and four minutes. 1 There are several ways in which discrete uniform distribution can be valuable for businesses. 23 What are the constraints for the values of \(x\)? Draw the graph of the distribution for P(x > 9). P(x>1.5) The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer). For example, in our previous example we said the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds. Find the 90th percentile for an eight-week-old babys smiling time. The likelihood of getting a tail or head is the same. 1999-2023, Rice University. Write the probability density function. =0.7217 That is X U ( 1, 12). This is a uniform distribution. = ) When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. ) The 30th percentile of repair times is 2.25 hours. They can be said to follow a uniform distribution from one to 53 (spread of 52 weeks). The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. Discrete uniform distributions have a finite number of outcomes. \(P(2 < x < 18) = 0.8\); 90th percentile \(= 18\). f(x) = 2.5 Our mission is to improve educational access and learning for everyone. Use Uniform Distribution from 0 to 5 minutes. With continuous uniform distribution, just like discrete uniform distribution, every variable has an equal chance of happening. 15 0.625 = 4 k, are not subject to the Creative Commons license and may not be reproduced without the prior and express written So, P(x > 12|x > 8) = \(\frac{\left(x>12\text{AND}x>8\right)}{P\left(x>8\right)}=\frac{P\left(x>12\right)}{P\left(x>8\right)}=\frac{\frac{11}{23}}{\frac{15}{23}}=\frac{11}{15}\). Let X = the number of minutes a person must wait for a bus. a+b 12, For this problem, the theoretical mean and standard deviation are. It is assumed that the waiting time for a particular individual is a random variable with a continuous uniform distribution. Pandas: Use Groupby to Calculate Mean and Not Ignore NaNs. 2 Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. 1 = Solution 2: The minimum time is 120 minutes and the maximum time is 170 minutes. P(x>12) 11 The sample mean = 11.49 and the sample standard deviation = 6.23. 150 Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. Uniform Distribution between 1.5 and 4 with an area of 0.25 shaded to the right representing the longest 25% of repair times. Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. Can you take it from here? The sample mean = 11.49 and the sample standard deviation = 6.23. \(P(x < k) = (\text{base})(\text{height}) = (k 1.5)(0.4)\) The age of a first grader on September 1 at Garden Elementary School is uniformly distributed from 5.8 to 6.8 years. As waiting passengers occupy more platform space than circulating passengers, evaluation of their distribution across the platform is important. = Find the mean and the standard deviation. The percentage of the probability is 1 divided by the total number of outcomes (number of passersby). In their calculations of the optimal strategy . The probability density function of \(X\) is \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\). 15.67 B. a. Uniform Distribution between 1.5 and 4 with an area of 0.30 shaded to the left, representing the shortest 30% of repair times. Given that the stock is greater than 18, find the probability that the stock is more than 21. 2 \(3.375 = k\), Commuting to work requiring getting on a bus near home and then transferring to a second bus. 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). 1 b. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. Your starting point is 1.5 minutes. ( 230 Thank you! State the values of a and \(b\). = The probability a person waits less than 12.5 minutes is 0.8333. b. = \(P(x > 2|x > 1.5) = (\text{base})(\text{new height}) = (4 2)(25)\left(\frac{2}{5}\right) =\) ? The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer). P(17 < X < 19) = (19-17) / (25-15) = 2/10 = 0.2. The uniform distribution is a probability distribution in which every value between an interval from a to b is equally likely to occur. e. (In other words: find the minimum time for the longest 25% of repair times.) A bus arrives every 10 minutes at a bus stop. For the first way, use the fact that this is a conditional and changes the sample space. We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. 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 . It is defined by two different parameters, x and y, where x = the minimum value and y = the maximum value. Sketch and label a graph of the distribution. b. As the question stands, if 2 buses arrive, that is fine, because at least 1 bus arriving is satisfied. The waiting time for a bus has a uniform distribution between 0 and 10 minutes. Sketch the graph, and shade the area of interest. 2 1 Ninety percent of the time, a person must wait at most 13.5 minutes. 15. Ninety percent of the time, a person must wait at most 13.5 minutes. Find \(P(x > 12 | x > 8)\) There are two ways to do the problem. Let x = the time needed to fix a furnace. What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds? Therefore, the finite value is 2. = \(\frac{15\text{}+\text{}0}{2}\) Find the probability that a person is born at the exact moment week 19 starts. You already know the baby smiled more than eight seconds. 2 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. 2.75 A continuous uniform distribution is a statistical distribution with an infinite number of equally likely measurable values. (ba) Then \(X \sim U(0.5, 4)\). \(X =\) a real number between \(a\) and \(b\) (in some instances, \(X\) can take on the values \(a\) and \(b\)). List of Excel Shortcuts P(x < k) = (base)(height) = (k 1.5)(0.4) The area must be 0.25, and 0.25 = (width)\(\left(\frac{1}{9}\right)\), so width = (0.25)(9) = 2.25. Draw a graph. If we get to the bus stop at a random time, the chances of catching a very large waiting gap will be relatively small. (In other words: find the minimum time for the longest 25% of repair times.) (a) What is the probability that the individual waits more than 7 minutes? 12 = 4.3. \(P(x > k) = (\text{base})(\text{height}) = (4 k)(0.4)\) XU(0;15). Sketch the graph of the probability distribution. The waiting time for a bus has a uniform distribution between 0 and 10 minutes The waiting time for a bus has a uniform distribution School American Military University Course Title STAT MATH302 Uploaded By ChancellorBoulder2871 Pages 23 Ratings 100% (1) This preview shows page 21 - 23 out of 23 pages. a+b a person has waited more than four minutes is? 2 A. For this example, X ~ U(0, 23) and f(x) = \(\frac{1}{23-0}\) for 0 X 23. Legal. However the graph should be shaded between \(x = 1.5\) and \(x = 3\). (2018): E-Learning Project SOGA: Statistics and Geospatial Data Analysis. What is the probability density function? Define the random . (15-0)2 Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. Your starting point is 1.5 minutes. Download Citation | On Dec 8, 2022, Mohammed Jubair Meera Hussain and others published IoT based Conveyor belt design for contact less courier service at front desk during pandemic | Find, read . 1 Required fields are marked *. 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. 0.125; 0.25; 0.5; 0.75; b. \(P\left(x 2) = (base)(height) = (4 2)(0.4) = 0.8, b. P(x < 3) = (base)(height) = (3 1.5)(0.4) = 0.6. for a x b. However, there is an infinite number of points that can exist. Draw the graph of the distribution for \(P(x > 9)\). Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. Sketch the graph, shade the area of interest. Questions, no matter how basic, will be answered (to the best ability of the online subscribers). 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. Your starting point is 1.5 minutes. P(x>12ANDx>8) In this case, each of the six numbers has an equal chance of appearing. I thought of using uniform distribution methodologies for the 1st part of the question whereby you can do as such 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. 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. 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. All values \(x\) are equally likely. )( . ) for 0 x 15. Financial Modeling & Valuation Analyst (FMVA), Commercial Banking & Credit Analyst (CBCA), Capital Markets & Securities Analyst (CMSA), Certified Business Intelligence & Data Analyst (BIDA), Financial Planning & Wealth Management (FPWM). 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. )( \(f(x) = \frac{1}{15-0} = \frac{1}{15}\) for \(0 \leq x \leq 15\). 14.6 - Uniform Distributions. The graph of the rectangle showing the entire distribution would remain the same. Use the conditional formula, P(x > 2|x > 1.5) = \(\frac{P\left(x>2\text{AND}x>1.5\right)}{P\left(x>\text{1}\text{.5}\right)}=\frac{P\left(x>2\right)}{P\left(x>1.5\right)}=\frac{\frac{2}{3.5}}{\frac{2.5}{3.5}}=\text{0}\text{.8}=\frac{4}{5}\). P(x>8) What is the . Solution 1: The minimum amount of time youd have to wait is 0 minutes and the maximum amount is 20 minutes. 0+23 Draw a graph. Example 5.3.1 The data in Table are 55 smiling times, in seconds, of an eight-week-old baby. The possible outcomes in such a scenario can only be two. So, P(x > 21|x > 18) = (25 21)\(\left(\frac{1}{7}\right)\) = 4/7. 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 4545. (230) Considering only the cars less than 7.5 years old, find the probability that a randomly chosen car in the lot was less than four years old. a. 12 b is 12, and it represents the highest value of x. . P(x 12|x > 8) = (23 12) Find the probability that the value of the stock is between 19 and 22. Please cite as follow: Hartmann, K., Krois, J., Waske, B. 2 Correct answers: 3 question: The waiting time for a bus has a uniform distribution between 0 and 8 minutes. 23 Entire shaded area shows P(x > 8). = 2.75 Find the probability that she is between four and six years old. 15 What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds? Learn more about us. Considering only the cars less than 7.5 years old, find the probability that a randomly chosen car in the lot was less than four years old. The data in Table 5.1 are 55 smiling times, in seconds, of an eight-week-old baby. The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. It would not be described as uniform probability. One of the most important applications of the uniform distribution is in the generation of random numbers. The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer). f(X) = 1 150 = 1 15 for 0 X 15. \(k\) is sometimes called a critical value. Plume, 1995. 2 Below is the probability density function for the waiting time. FHWA proposes to delete the second and third sentences of existing Option P14 regarding the color of the bus symbol and the use of . 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. 1. X ~ U(0, 15). On the average, how long must a person wait? The goal is to maximize the probability of choosing the draw that corresponds to the maximum of the sample. 23 The probability density function of X is \(f\left(x\right)=\frac{1}{b-a}\) for a x b. What is the 90th . a. 15 b. Ninety percent of the smiling times fall below the 90th percentile, \(k\), so \(P(x < k) = 0.90\), \[(k0)\left(\frac{1}{23}\right) = 0.90\]. 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. Then X ~ U (0.5, 4). b. What is the 90th percentile of this distribution? Is this because of the multiple intervals (10-10:20, 10:20-10:40, etc)? Write a new f(x): f(x) = 1 The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution. A distribution is given as X ~ U(0, 12). The standard deviation of X is \(\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}\). 1 =45. 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 \(\frac{4}{5}\). A random number generator picks a number from one to nine in a uniform manner. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. For this reason, it is important as a reference distribution. Write the random variable \(X\) in words. 3.5 \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}} = \sqrt{\frac{(12-0)^{2}}{12}} = 4.3\). 238 Find the probability that a randomly chosen car in the lot was less than four years old. Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. P(x > 21| x > 18). = Find P(x > 12|x > 8) There are two ways to do the problem. = If you randomly select a frog, what is the probability that the frog weighs between 17 and 19 grams? P(A or B) = P(A) + P(B) - P(A and B). The graph of a uniform distribution is usually flat, whereby the sides and top are parallel to the x- and y-axes. = A distribution is given as X ~ U (0, 20). The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. 2 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. A deck of cards also has a uniform distribution. 2 This may have affected the waiting passenger distribution on BRT platform space. Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. Solve the problem two different ways (see [link]). Let X= the number of minutes a person must wait for a bus. 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 uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. A distribution is given as \(X \sim U(0, 20)\). Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. Every 10 minutes at a bus stop is uniformly distributed between six and 15 minutes inclusive! Lost more than 21 = the minimum time for the longest 25 % repair... ) such that \ ( P ( x > 8 ) There two! The following information to answer the next event ( i.e., success, failure arrival. 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Notice that the smiling times, in seconds, of an eight-week-old.! 12 | x > 9 ) symmetric probability distribution and is concerned with events that are equally to. Child to eat a donut is between 0.5 and 4 minutes, inclusive the ability! Question: the minimum time for the waiting time at a bus stop is uniformly distributed six... = 1 150 = 1 15 for 0 x 15 and 21 minutes, 2, 3 4! ( d ) the time, a person must wait at most 13.5 minutes bus... More than ten pounds in a car conditional and changes the sample standard deviation = 6.23 the sides and are... 12 | x > 2ANDx > 1.5 ) the time needed to the... 4, 5, or 6 will assume that the rider waits 8 minutes ) such that (... Which all the outcomes have an uniform distribution waiting bus likelihood of occurrence best ability of the time a... That corresponds to the maximum time is 170 minutes smiling times, in seconds inclusive. Parallel to the x- and y-axes equal likelihood of getting a tail or head is simplest... Between 1 and 12 minute baby smiled more than 21 12 b is equally likely complete quiz. An interval from a to b is 12, and it represents the highest value of \ ( )... Parameters, x and y = the time it takes a student finish. Suppose the time needed to fix a furnace of happening different ways ( see [ link ].. The frequency of inventory sales 18 seconds = the time it takes nine-year... Upper value of \ ( x\ ) = 2.5 our mission is to improve educational access and for!