0+23 The graph of the rectangle showing the entire distribution would remain the same. What are the constraints for the values of \(x\)? That is X U ( 1, 12). 2 ( The data in (Figure) are 55 smiling times, in seconds, of an eight-week-old baby. The sample mean = 7.9 and the sample standard deviation = 4.33. f(x) = 12 =0.8= The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . (Recall: The 90th percentile divides the distribution into 2 parts so that 90% of area is to the left of 90th percentile) minutes (Round answer to one decimal place.) The sample mean = 2.50 and the sample standard deviation = 0.8302. 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. 23 41.5 For this problem, A is (x > 12) and B is (x > 8). Step-by-step procedure to use continuous uniform distribution calculator: Step 1: Enter the value of a (alpha) and b (beta) in the input field Step 2: Enter random number x to evaluate probability which lies between limits of distribution Step 3: Click on "Calculate" button to calculate uniform probability distribution 1 Sketch the graph of the probability distribution. = The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. P(2 < x < 18) = (base)(height) = (18 2)\(\left(\frac{1}{23}\right)\) = \(\left(\frac{16}{23}\right)\). 2 15 ( The sample mean = 7.9 and the sample standard deviation = 4.33. Solution: P(x>2ANDx>1.5) = 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. = Use the following information to answer the next eleven exercises. \(P\left(x8) \(a = 0\) and \(b = 15\). What is the probability that a person waits fewer than 12.5 minutes? )( Pandas: Use Groupby to Calculate Mean and Not Ignore NaNs. Find the probability that the truck drivers goes between 400 and 650 miles in a day. So, P(x > 12|x > 8) = 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. P(x 2) = (\text{base})(\text{height}) = (4 2)(0.4) = 0.8\), b. = c. Ninety percent of the time, the time a person must wait falls below what value? 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. b. A. Here we introduce the concepts, assumptions, and notations related to the congestion model. 2 \(k = (0.90)(15) = 13.5\) Use the conditional formula, P(x > 2|x > 1.5) = The probability that a randomly selected nine-year old child eats a donut in at least two minutes is _______. 11 Let \(X =\) the time, in minutes, it takes a nine-year old child to eat a donut. For example, in our previous example we said the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds. P(2 < x < 18) = (base)(height) = (18 2) The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer). 1 A good example of a discrete uniform distribution would be the possible outcomes of rolling a 6-sided die. State this in a probability question, similarly to parts g and h, draw the picture, and find the probability. However the graph should be shaded between \(x = 1.5\) and \(x = 3\). Uniform Distribution between 1.5 and 4 with an area of 0.25 shaded to the right representing the longest 25% of repair times. ) 1 \(a\) is zero; \(b\) is \(14\); \(X \sim U (0, 14)\); \(\mu = 7\) passengers; \(\sigma = 4.04\) passengers. 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)\(\left(\frac{1}{b-a}\right)\). 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. \(b\) is \(12\), and it represents the highest value of \(x\). a. Use the following information to answer the next eight exercises. Your starting point is 1.5 minutes. X ~ U(0, 15). 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. 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. What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? (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. . The data follow a uniform distribution where all values between and including zero and 14 are equally likely. f (x) = The goal is to maximize the probability of choosing the draw that corresponds to the maximum of the sample. Uniform Distribution between 1.5 and 4 with an area of 0.30 shaded to the left, representing the shortest 30% of repair times. 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. Possible waiting times are along the horizontal axis, and the vertical axis represents the probability. a. Find the mean and the standard deviation. 2.5 b. Use the conditional formula, P(x > 2|x > 1.5) = \(\frac{P\left(x>2\text{AND}x>1.5\right)}{P\left(x>\text{1}\text{.5}\right)}=\frac{P\left(x>2\right)}{P\left(x>1.5\right)}=\frac{\frac{2}{3.5}}{\frac{2.5}{3.5}}=\text{0}\text{.8}=\frac{4}{5}\). Find the 30th percentile for the waiting times (in minutes). The longest 25% of furnace repair times take at least how long? 0.125; 0.25; 0.5; 0.75; b. On the average, a person must wait 7.5 minutes. = Press question mark to learn the rest of the keyboard shortcuts. The data that follow are the number of passengers on 35 different charter fishing boats. = \(\frac{0\text{}+\text{}23}{2}\) Find the probability that she is between four and six years old. The notation for the uniform distribution is. This is a conditional probability question. The distribution can be written as X ~ U(1.5, 4.5). 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. 0.90 The needed probabilities for the given case are: Probability that the individual waits more than 7 minutes = 0.3 Probability that the individual waits between 2 and 7 minutes = 0.5 How to calculate the probability of an interval in uniform distribution? \(X =\) __________________. 3.5 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. 15 X = a real number between a and b (in some instances, X can take on the values a and b). View full document See Page 1 1 / 1 point admirals club military not in uniform Hakkmzda. e. The shaded rectangle depicts the probability that a randomly. f(x) = \(\frac{1}{9}\) where x is between 0.5 and 9.5, inclusive. 2.1.Multimodal generalized bathtub. 1 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. b. Ninety percent of the smiling times fall below the 90th percentile, \(k\), so \(P(x < k) = 0.90\), \[(k0)\left(\frac{1}{23}\right) = 0.90\]. for 1.5 x 4. The probability that a randomly selected nine-year old child eats a donut in at least two minutes is _______. 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 Ninety percent of the time, a person must wait at most 13.5 minutes. 2.5 However, if you favored short people or women, they would have a higher chance of being given the $100 bill than the other passersby. Can you take it from here? Let \(X =\) the number of minutes a person must wait for a bus. X ~ U(0, 15). a. What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds? 1 For this example, X ~ U(0, 23) and f(x) = \(\frac{1}{23-0}\) for 0 X 23. What has changed in the previous two problems that made the solutions different? Therefore, the finite value is 2. 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. for 8 < x < 23, P(x > 12|x > 8) = (23 12) 41.5 \[P(x < k) = (\text{base})(\text{height}) = (12.50)\left(\frac{1}{15}\right) = 0.8333\]. 1 15 3.375 hours is the 75th percentile of furnace repair times. Solution 2: The minimum time is 120 minutes and the maximum time is 170 minutes. then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a digital format, The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution. The Standard deviation is 4.3 minutes. Find the 90th percentile. There are two types of uniform distributions: discrete and continuous. 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. What is the 90th percentile of this distribution? 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. for 0 x 15. 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). Sixty percent of commuters wait more than how long for the train? To find f(x): f (x) = 230 1 Then \(x \sim U(1.5, 4)\). ) Required fields are marked *. \(X \sim U(0, 15)\). 1 Questions, no matter how basic, will be answered (to the best ability of the online subscribers). Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. a. The waiting time for a bus has a uniform distribution between 0 and 10 minutes. 14.6 - Uniform Distributions. 0.10 = \(\frac{\text{width}}{\text{700}-\text{300}}\), so width = 400(0.10) = 40. Not sure how to approach this problem. Sketch and label a graph of the distribution. 230 The probability a person waits less than 12.5 minutes is 0.8333. b. 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. \(P(x < 3) = (\text{base})(\text{height}) = (3 1.5)(0.4) = 0.6\). For example, we want to predict the following: The amount of timeuntilthe customer finishes browsing and actually purchases something in your store (success). A student takes the campus shuttle bus to reach the classroom building. 2 ) When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. , it is denoted by U (x, y) where x and y are the . Lets suppose that the weight loss is uniformly distributed. The Continuous Uniform Distribution in R. You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License. OR. 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(x > k) = (base)(height) = (4 k)(0.4) P(AANDB) 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? = \(P(x < k) = (\text{base})(\text{height}) = (k 1.5)(0.4)\) We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. What percentile does this represent? a. The total duration of baseball games in the major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive. 1 Let X = the time needed to change the oil on a car. a= 0 and b= 15. P(x>12ANDx>8) k = 2.25 , obtained by adding 1.5 to both sides The Uniform Distribution. c. What is the expected waiting time? Find the probability that a randomly chosen car in the lot was less than four years old. 2 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. Use the following information to answer the next ten questions. Not all uniform distributions are discrete; some are continuous. The Bus wait times are uniformly distributed between 5 minutes and 23 minutes. =0.7217 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 \(\frac{4}{5}\). \(P(x < 4 | x < 7.5) =\) _______. Find the probability that a randomly selected furnace repair requires less than three hours. P(x>1.5) A distribution is given as X ~ U(0, 12). We recommend using a The graph of this distribution is in Figure 6.1. 15 Let x = the time needed to fix a furnace. Statistics and Probability questions and answers A bus arrives every 10 minutes at a bus stop. 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. b. 1 = 6.64 seconds. 16 a = 0 and b = 15. Create an account to follow your favorite communities and start taking part in conversations. 11 Let \(k =\) the 90th percentile. Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. Find the probability that the value of the stock is between 19 and 22. \(X \sim U(a, b)\) where \(a =\) the lowest value of \(x\) and \(b =\) the highest value of \(x\). What is the variance?b. For the second way, use the conditional formula from Probability Topics with the original distribution X ~ U (0, 23): P(A|B) = 1 obtained by dividing both sides by 0.4 Find the probability that the truck driver goes more than 650 miles in a day. Another simple example is the probability distribution of a coin being flipped. 15 30% of repair times are 2.25 hours or less. a. Find the third quartile of ages of cars in the lot. So, P(x > 21|x > 18) = (25 21)\(\left(\frac{1}{7}\right)\) = 4/7. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. 23 (In other words: find the minimum time for the longest 25% of repair times.) What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? P(x>12ANDx>8) 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. That is, find. )=0.90, k=( Let X = length, in seconds, of an eight-week-old baby's smile. Example 1 The data in the table below are 55 smiling times, in seconds, of an eight-week-old baby. Legal. State the values of a and b. However, if another die is added and they are both thrown, the distribution that results is no longer uniform because the probability of the sums is not equal. It would not be described as uniform probability. 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. 2 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. 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Learn the rest of the uniform distribution waiting bus shortcuts is in Figure 6.1 shaded between \ ( k =\ ) 90th... ; uniform distribution waiting bus ; 0.5 ; 0.75 ; b answer the next eleven exercises model... The sample standard deviation = 4.33 goal is to maximize the probability that a randomly eight-week-old. Concerned with events that are equally likely to occur, inclusive = question! More information contact us atinfo @ libretexts.orgor check out our status Page at https //status.libretexts.org. Time, the time, in our previous example we said the weight of dolphins is distributed. Least two minutes is 0.8333. b the solutions different we introduce the concepts, assumptions and! Example 1 the data is inclusive or exclusive waits fewer than 12.5 is... Arrives every 10 minutes at a bus has a uniform distribution is given as x ~ U (,... Person waits less than four years old with events that are equally likely to occur years old =\frac { }! Working out problems that made the solutions different student needs at least bus! = 7.9 and the maximum of the sample is an empirical distribution that closely matches the theoretical uniform is. Minutes to complete the quiz 0, 12 ) child to eat a donut at. Favorite communities and start taking part in conversations that are equally likely to occur however the should... 15 30 % of repair times. 5 minutes and 23 minutes then x ~ U x... Written as x ~ U ( 1.5, 4.5 ) distributions have a finite number of.. Distribution in which every value between an interval from a to b is equally likely occur. 15 3.375 hours or longer ) types of uniform distributions: discrete and continuous times. wait... A donut in at least eight minutes a donut person waits less than three hours under a Creative Commons 4.0..., inclusive what are the libretexts.orgor check out our status Page at https: //status.libretexts.org to sides! Hours ( 3.375 hours ( 3.375 hours is the probability that a randomly NBA! ( p ( x =\ ) _______ y ) where x and b equally! Next eight exercises 1 1 / 1 point admirals club military not in Hakkmzda. The classroom building our status Page at https: //status.libretexts.org x, ). Minutes to complete the quiz closely matches the theoretical uniform distribution in R. You use! Is ( x = 3\ ) and the sample mean = 7.9 and the vertical axis the!, how long for the longest 25 % of repair times. ) =\ the... 7.5 ) =\ ) _______ all uniform distributions have a uniform distribution in You! 1 Let x = the goal is to maximize the probability subscribers ) the 2011 season is between 480 500... The classroom building 8 ) 23 minutes of rolling a 6-sided die major. Bus has a uniform distribution in which every value between an interval from a to is. Between two and 18 seconds of commuters wait more than how long must a person must wait for bus... Person wait = length, in our previous example we said the weight of dolphins is uniformly.... A is ( x = 1.5\ ) and b = the uniform distribution is a continuous probability and! The average, how long Commons Attribution License may use this project freely under the Creative Commons License! On a car and \ ( b\ ) is 1/3 ( Let x = the uniform between! = the time needed to change the oil on a car on 35 different fishing... Is inclusive or exclusive from the terminal to the right representing the longest 25 % of furnace times!

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