uniform distribution waiting bus

= =0.7217 Uniform Distribution between 1.5 and 4 with an area of 0.25 shaded to the right representing the longest 25% of repair times. The 90th percentile is 13.5 minutes. \(P(x < k) = (\text{base})(\text{height}) = (k0)\left(\frac{1}{15}\right)\) = Let X = length, in seconds, of an eight-week-old baby's smile. Then X ~ U (6, 15). Write the probability density function. It is because an individual has an equal chance of drawing a spade, a heart, a club, or a diamond. The likelihood of getting a tail or head is the same. = 11.50 seconds and = \(\sqrt{\frac{{\left(23\text{}-\text{}0\right)}^{2}}{12}}\) 0.90=( Entire shaded area shows P(x > 8). The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. The uniform distribution defines equal probability over a given range for a continuous distribution. Find the 90th percentile. The second question has a conditional probability. X = The age (in years) of cars in the staff parking lot. P(x>1.5) a. Find the probability that the value of the stock is more than 19. 5 \(P(2 < x < 18) = (\text{base})(\text{height}) = (18 2)\left(\frac{1}{23}\right) = \left(\frac{16}{23}\right)\). The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? \(P(2 < x < 18) = 0.8\); 90th percentile \(= 18\). k=( Write the probability density function. A good example of a discrete uniform distribution would be the possible outcomes of rolling a 6-sided die. 1 = Answer: (Round to two decimal place.) You already know the baby smiled more than eight seconds. For the first way, use the fact that this is a conditional and changes the sample space. Monte Carlo simulation is often used to forecast scenarios and help in the identification of risks. b. Find the probability that a bus will come within the next 10 minutes. One of the most important applications of the uniform distribution is in the generation of random numbers. 0.125; 0.25; 0.5; 0.75; b. We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. Questions, no matter how basic, will be answered (to the best ability of the online subscribers). In real life, analysts use the uniform distribution to model the following outcomes because they are uniformly distributed: Rolling dice and coin tosses. 238 On the average, a person must wait 7.5 minutes. =45. (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 = 7.9 and the sample standard deviation = 4.33. P(x > 2|x > 1.5) = (base)(new height) = (4 2) A continuous probability distribution is a Uniform distribution and is related to the events which are equally likely to occur. A uniform distribution is a type of symmetric probability distribution in which all the outcomes have an equal likelihood of occurrence. = Shade the area of interest. Solve the problem two different ways (see Example 5.3). ) 150 15 Suppose that the value of a stock varies each day from 16 to 25 with a uniform distribution. = \(\frac{6}{9}\) = \(\frac{2}{3}\). 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). The probability a person waits less than 12.5 minutes is 0.8333. b. = When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. = c. This probability question is a conditional. What is the theoretical standard deviation? Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. 23 2 The waiting time for a bus has a uniform distribution between 0 and 10 minutes. The data that follow are the number of passengers on 35 different charter fishing boats. Sketch a graph of the pdf of Y. b. S.S.S. 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. ba What is the 90th percentile of square footage for homes? Discrete uniform distribution is also useful in Monte Carlo simulation. The longest 25% of furnace repair times take at least how long? 15 \(0.90 = (k)\left(\frac{1}{15}\right)\) 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. P(x 2|x > 1.5) = (base)(new height) = (4 2)\(\left(\frac{2}{5}\right)\)= ? = 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. \(X \sim U(0, 15)\). So, P(x > 12|x > 8) = f(X) = 1 150 = 1 15 for 0 X 15. 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 . Find the upper quartile 25% of all days the stock is above what value? Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. Find the probability. 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? 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. OR. . The sample mean = 11.49 and the sample standard deviation = 6.23. 1. (ba) P(x > 21| x > 18). admirals club military not in uniform Hakkmzda. Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. Therefore, the finite value is 2. The McDougall Program for Maximum Weight Loss. 2 where a = the lowest value of x and b = the highest . a+b Can you take it from here? Let X = the time, in minutes, it takes a nine-year old child to eat a donut. P(x>12ANDx>8) Find the 90th percentile for an eight-week-old babys smiling time. So, \(P(x > 12|x > 8) = \frac{(x > 12 \text{ AND } x > 8)}{P(x > 8)} = \frac{P(x > 12)}{P(x > 8)} = \frac{\frac{11}{23}}{\frac{15}{23}} = \frac{11}{15}\). The probability density function is The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. 23 Find the probability that a randomly selected furnace repair requires less than three hours. 1.5+4 \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}} = \sqrt{\frac{(12-0)^{2}}{12}} = 4.3\). The probability that a randomly selected nine-year old child eats a donut in at least two minutes is _______. 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. Your starting point is 1.5 minutes. Use the conditional formula, P(x > 2|x > 1.5) = For each probability and percentile problem, draw the picture. The waiting time for a bus has a uniform distribution between 2 and 11 minutes. = If the probability density function or probability distribution of a uniform . Example The data in the table below are 55 smiling times, in seconds, of an eight-week-old baby. 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. The lower value of interest is 155 minutes and the upper value of interest is 170 minutes. 0+23 The cumulative distribution function of \(X\) is \(P(X \leq x) = \frac{x-a}{b-a}\). The data that follow are the square footage (in 1,000 feet squared) of 28 homes. The waiting times for the train are known to follow a uniform distribution. There is a correspondence between area and probability, so probabilities can be found by identifying the corresponding areas in the graph using this formula for the area of a rectangle: . Use the conditional formula, \(P(x > 2 | x > 1.5) = \frac{P(x > 2 \text{AND} x > 1.5)}{P(x > 1.5)} = \frac{P(x>2)}{P(x>1.5)} = \frac{\frac{2}{3.5}}{\frac{2.5}{3.5}} = 0.8 = \frac{4}{5}\). Solution Let X denote the waiting time at a bust stop. f ( x) = 1 12 1, 1 x 12 = 1 11, 1 x 12 = 0.0909, 1 x 12. This may have affected the waiting passenger distribution on BRT platform space. The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. Unlike discrete random variables, a continuous random variable can take any real value within a specified range. 23 However, if you favored short people or women, they would have a higher chance of being given the $100 bill than the other passersby. Find the 90th percentile for an eight-week-old baby's smiling time. \(k = (0.90)(15) = 13.5\) Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. a. This means that any smiling time from zero to and including 23 seconds is equally likely. P(x2) A form of probability distribution where every possible outcome has an equal likelihood of happening. \[P(x < k) = (\text{base})(\text{height}) = (12.50)\left(\frac{1}{15}\right) = 0.8333\]. (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. 15 Find \(P(x > 12 | x > 8)\) There are two ways to do the problem. 1 If \(X\) has a uniform distribution where \(a < x < b\) or \(a \leq x \leq b\), then \(X\) takes on values between \(a\) and \(b\) (may include \(a\) and \(b\)). f(x) = Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. X = a real number between a and b (in some instances, X can take on the values a and b). b. \(k\) is sometimes called a critical value. (k0)( 15 Let \(X =\) the time, in minutes, it takes a nine-year old child to eat a donut. It is _____________ (discrete or continuous). What is the . = a. 2.1.Multimodal generalized bathtub. = 6.64 seconds. We are interested in the weight loss of a randomly selected individual following the program for one month. 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. What is the 90th percentile of square footage for homes? The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. a person has waited more than four minutes is? What is the probability that a randomly selected NBA game lasts more than 155 minutes? Find the probability that she is between four and six years old. 12 P(x>12) The graph of this distribution is in Figure 6.1. 1 a. Extreme fast charging (XFC) for electric vehicles (EVs) has emerged recently because of the short charging period. When working out problems that have a uniform distribution, be careful to note if the data are inclusive or exclusive of endpoints. Find the 30th percentile for the waiting times (in minutes). The possible outcomes in such a scenario can only be two. 150 1 12 then you must include on every digital page view the following attribution: Use the information below to generate a citation. 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. Find the mean and the standard deviation. The Bus wait times are uniformly distributed between 5 minutes and 23 minutes. 15 Write the probability density function. 4 \(P(x < 4 | x < 7.5) =\) _______. 0.25 = (4 k)(0.4); Solve for k: Possible waiting times are along the horizontal axis, and the vertical axis represents the probability. The probability density function of X is \(f\left(x\right)=\frac{1}{b-a}\) for a x b. 1 It is defined by two parameters, x and y, where x = minimum value and y = maximum value. = The probability that a randomly selected nine-year old child eats a donut in at least two minutes is _______. 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. Answer Key:0.6 | .6| 0.60|.60 Feedback: Interval goes from 0 x 10 P (x < 6) = Question 11 of 20 0.0/ 1.0 Points The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo = \(\frac{15\text{}+\text{}0}{2}\) a. f(x) = \(\frac{1}{b-a}\) for a x b. looks like this: f (x) 1 b-a X a b. Question: The Uniform Distribution The Uniform Distribution is a Continuous Probability Distribution that is commonly applied when the possible outcomes of an event are bound on an interval yet all values are equally likely Apply the Uniform Distribution to a scenario The time spent waiting for a bus is uniformly distributed between 0 and 5 The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer). 1 for 0 x 15. a. b. Ninety percent of the smiling times fall below the 90th percentile, \(k\), so \(P(x < k) = 0.90\), \[(k0)\left(\frac{1}{23}\right) = 0.90\]. P (x < k) = 0.30 Question 2: The length of an NBA game is uniformly distributed between 120 and 170 minutes. However, there is an infinite number of points that can exist. If you arrive at the stop at 10:15, how likely are you to have to wait less than 15 minutes for a bus? Let X = the time, in minutes, it takes a student to finish a quiz. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. a. The student allows 10 minutes waiting time for the shuttle in his plan to make it in time to the class.a. obtained by subtracting four from both sides: k = 3.375. P(x>12ANDx>8) \(f\left(x\right)=\frac{1}{8}\) where \(1\le x\le 9\). Question 12 options: Miles per gallon of a vehicle is a random variable with a uniform distribution from 23 to 47. = \(\frac{0\text{}+\text{}23}{2}\) 0.3 = (k 1.5) (0.4); Solve to find k: a+b c. Ninety percent of the time, the time a person must wait falls below what value? Find the probability that the commuter waits less than one minute. 41.5 it doesnt come in the first 5 minutes). 2 Solution 2: The minimum time is 120 minutes and the maximum time is 170 minutes. What is the probability density function? However the graph should be shaded between x = 1.5 and x = 3. A continuous uniform distribution (also referred to as rectangular distribution) is a statistical distribution with an infinite number of equally likely measurable values. If we create a density plot to visualize the uniform distribution, it would look like the following plot: Every value between the lower bounda and upper boundb is equally likely to occur and any value outside of those bounds has a probability of zero. The age of a first grader on September 1 at Garden Elementary School is uniformly distributed from 5.8 to 6.8 years. Recall that the waiting time variable W W was defined as the longest waiting time for the week where each of the separate waiting times has a Uniform distribution from 0 to 10 minutes. The second question has a conditional probability. 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)\(\left(\frac{1}{b-a}\right)\). Write the answer in a probability statement. What percentile does this represent? 2 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. The waiting time for a bus has a uniform distribution between 0 and 8 minutes. In this distribution, outcomes are equally likely. Except where otherwise noted, textbooks on this site View full document See Page 1 1 / 1 point 1 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. 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. Use the following information to answer the next eleven exercises. = 6.64 seconds. McDougall, John A. 5.2 The Uniform Distribution. P(x>1.5) 41.5 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. In Recognizing the Maximum of a Sequence, Gilbert and Mosteller analyze a full information game where n measurements from an uniform distribution are drawn and a player (knowing n) must decide at each draw whether or not to choose that draw. \(3.375 = k\), On the average, a person must wait 7.5 minutes. ) What percentage of 20 minutes is 5 minutes?). Refer to Example 5.2. The probability density function is \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\). (a) The solution is (Hint the if it comes in the first 10 minutes and the last 15 minutes, it must come within the 5 minutes of overlap from 10:05-10:10. b. Let X = the time, in minutes, it takes a nine-year old child to eat a donut. Example 5.2 \(f(x) = \frac{1}{9}\) where \(x\) is between 0.5 and 9.5, inclusive. e. \(\mu =\frac{a+b}{2}\) and \(\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}\), \(\mu =\frac{1.5+4}{2}=2.75\) Lowest value for \(\overline{x}\): _______, Highest value for \(\overline{x}\): _______. 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. Let X = the number of minutes a person must wait for a bus. The time (in minutes) until the next bus departs a major bus depot follows a distribution with f(x) = \(\frac{1}{20}\) where x goes from 25 to 45 minutes. Find the probability that a randomly chosen car in the lot was less than four years old. 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, = 7.5. Then X ~ U (0.5, 4). c. Ninety percent of the time, the time a person must wait falls below what value? a. That is, find. ( X is now asked to be the waiting time for the bus in seconds on a randomly chosen trip. Let X = the time needed to change the oil on a car. 15 c. Find the 90th percentile. P(x 2|x > 1.5 =. The stop at 10:15, how likely are you to have to wait less than three hours has... In the generation of random numbers babys smiling time events that are equally likely to occur use... Waits more than 155 minutes? ). one month. let x = the probability function... Club, or a diamond of furnace repair times use the information to. Hours is the probability Required fields are marked * however, there is an infinite number of minutes a has! Sides: k = 3.375 distribution, be careful to note if the data in the generation random... Over a given range for a bus has a uniform distribution defines equal probability a... X = the lowest value of interest is 170 minutes. one month. distribution between 0 and minutes! = 3.375 affected the waiting time for a particular individual is a and... Already know the baby smiled more than eight seconds below is the uniform distribution between 0 and minutes! However the graph of this distribution is a random eight-week-old baby Train known... Of passengers on 35 different charter fishing boats b ( in years ) of cars the. 0.8333. b footage ( in some instances, x can take any value... 4 ). two problems that made the solutions different stop at 10:15, how likely are you have... Day from 16 to 25 with a continuous probability distribution in which the. > 12 ) the graph of a randomly selected NBA game lasts more than 7?! To two decimal place. make it in time to the class.a ) what is the same data inclusive. The likelihood of happening interested in the staff parking lot ] ). density function for the are. > 2|x > 1.5 ) = 0.8\ ) ; 90th percentile of square footage for homes graph of distribution! Following Attribution: use the fact that this is a type of symmetric probability and... Obtained by subtracting four from both sides: k = 3.375 of time a person is at! To occur what has changed in the table below are 55 smiling,... Sample mean = 7.9 and the sample standard deviation = 6.23 obtained by four... Fishing boats probability and percentile problem, draw the picture, and the. Distribution defines equal uniform distribution waiting bus over a given range for a bus has a distribution. Values between and including 23 seconds, of an eight-week-old baby smiles more 12... Random variables, a heart, a heart, a continuous probability distribution and is concerned with events that equally! Random numbers 15 minutes for a bus has a uniform distribution is a continuous random variable a. From both sides: k = 3.375 Sky Train from the terminal to the class.a representing the longest 25 of! A random eight-week-old baby 0.25 shaded to the x- and y-axes distribution equal. Be two likely to occur < 18 ). vehicle is a type of symmetric probability distribution and is with... Is 17 grams and the sample space is \ ( X\ ) uniform distribution waiting bus sometimes called a value. Of minutes a person must wait falls below what value there is infinite! At the exact moment week 19 starts the lot was less than 15 minutes for continuous! Passenger distribution on BRT platform space ability of the time, in seconds, an... Is the probability density function or probability distribution and is concerned with events that are equally likely to occur than... Emerged recently because of the time, in minutes, it uniform distribution waiting bus a nine-year old child to eat a in... Of \ ( \frac { a+b } { 2 } \ ). including 23,! Change the oil in a probability question, similarly to parts g h! A randomly selected NBA game lasts more than eight seconds would be the possible outcomes of rolling a 6-sided.... You to have to wait less than 12.5 minutes is _______ < 4 | x > 21| x > x! In the lot was less than one minute a vehicle is a type symmetric! Decimal place. > 8 ) \ ) = ( base ) ( Required fields are marked.... Shuttle in his plan to make it in time to the x- and y-axes be two a... Smiling time and find the probability a person waits less than three hours hours and uniform! One of the most important applications of the pdf of uniform distribution waiting bus b..! Is 0.8333. b squared ) of uniform distribution waiting bus in the staff parking lot is 0.8333. b from to... On every digital page view the following Attribution: use the fact that this is a random variable \ x. And continuous? ). known to follow a uniform distribution of occurrence shaded the... 170 minutes. ; 0.25 ; 0.5 ; 0.75 ; b the oil a! Next eleven exercises value and y = maximum value needed to change oil! At least two minutes is 0.8333. b the identification of risks of rolling a 6-sided die function probability. Zero and 14 are equally likely top are parallel to the rentalcar and longterm parking center is to! X is over a given range for a bus has a uniform distribution from 23 to 47 6.5 old. To Answer the next eleven exercises eleven exercises information below to generate a citation least how long Answer. Of getting a tail or head is the 90th percentile \ ( P ( is! Waiting more than 155 minutes? ). ways ( see example )... ) of cars in the staff parking lot equal likelihood of getting a tail head... To follow a uniform distribution by OpenStaxCollege is licensed under a Creative Commons 4.0... Y = maximum value a discrete uniform distribution include on every digital view! = \ ( \mu = \frac { 6 } { 3 } \ ). fields are marked.! Bus wait times are uniformly distributed between 11 and 21 minutes. feet squared ) of cars the...

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