OR. Draw a graph. What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? b. Answer: a. The Standard deviation is 4.3 minutes. 15 P(X > 19) = (25 19) \(\left(\frac{1}{9}\right)\) Find the probability that she is over 6.5 years old. For this example, x ~ U(0, 23) and f(x) = c. Find the 90th percentile. A good example of a discrete uniform distribution would be the possible outcomes of rolling a 6-sided die. P(x>1.5) Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. obtained by dividing both sides by 0.4 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 19) = (19-17) / (25-15) = 2/10 =, How to Plot an Exponential Distribution in R. Your email address will not be published. (a) The probability density function of X is. Sketch the graph, shade the area of interest. I'd love to hear an explanation for these answers when you get one, because they don't make any sense to me. We recommend using a That is X U ( 1, 12). 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. 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: 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. The second question has a conditional probability. Uniform Distribution between 1.5 and 4 with an area of 0.25 shaded to the right representing the longest 25% of repair times. One of the most important applications of the uniform distribution is in the generation of random numbers. What does this mean? P(x>8) \(P(x > k) = (\text{base})(\text{height}) = (4 k)(0.4)\) The longest 25% of furnace repair times take at least how long? ) (a) What is the probability that the individual waits more than 7 minutes? 23 A uniform distribution is a type of symmetric probability distribution in which all the outcomes have an equal likelihood of occurrence. 5 Let X = length, in seconds, of an eight-week-old baby's smile. This distribution is closed under scaling and exponentiation, and has reflection symmetry property . X ~ U(0, 15). \(k\) is sometimes called a critical value. \(f(x) = \frac{1}{4-1.5} = \frac{2}{5}\) for \(1.5 \leq x \leq 4\). The data that follow are the square footage (in 1,000 feet squared) of 28 homes. The waiting time for a bus has a uniform distribution between 0 and 10 minutes. 2 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 15 Find \(P(x > 12 | x > 8)\) There are two ways to do the problem. Then X ~ U (0.5, 4). 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. View full document See Page 1 1 / 1 point 15 Find the mean and the standard deviation. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. 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 the first way, use the fact that this is a conditional and changes the sample space. = Best Buddies Turkey Ekibi; Videolar; Bize Ulan; admirals club military not in uniform 27 ub. = 7.5. 2 = ) Below is the probability density function for the waiting time. The probability of drawing any card from a deck of cards. Press J to jump to the feed. 2.75 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. Find the value \(k\) such that \(P(x < k) = 0.75\). \(X\) is continuous. For the second way, use the conditional formula from Probability Topics with the original distribution \(X \sim U(0, 23)\): \(P(\text{A|B}) = \frac{P(\text{A AND B})}{P(\text{B})}\). 11 Your probability of having to wait any number of minutes in that interval is the same. Question 3: The weight of a certain species of frog is uniformly distributed between 15 and 25 grams. 2.1.Multimodal generalized bathtub. Learn more about us. Statistics and Probability questions and answers A bus arrives every 10 minutes at a bus stop. 0.3 = (k 1.5) (0.4); Solve to find k: 15 Find the 90th percentile for an eight-week-old babys smiling time. 3 buses will arrive at the the same time (i.e. The number of values is finite. What is the probability that the rider waits 8 minutes or less? You must reduce the sample space. It is because an individual has an equal chance of drawing a spade, a heart, a club, or a diamond. Introduction to Statistics is our premier online video course that teaches you all of the topics covered in introductory statistics. \(3.375 = k\), The notation for the uniform distribution is. a+b 2.5 = P(x1.5) The probability a person waits less than 12.5 minutes is 0.8333. b. \(P(x < 3) = (\text{base})(\text{height}) = (3 1.5)(0.4) = 0.6\). A fireworks show is designed so that the time between fireworks is between one and five seconds, and follows a uniform distribution. f ( x) = 1 12 1, 1 x 12 = 1 11, 1 x 12 = 0.0909, 1 x 12. 30% of repair times are 2.5 hours or less. 15 Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. = P(x>12) c. Ninety percent of the time, the time a person must wait falls below what value? 11 c. This probability question is a conditional. Lowest value for \(\overline{x}\): _______, Highest value for \(\overline{x}\): _______. 16 a. 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. Public transport systems have been affected by the global pandemic Coronavirus disease 2019 (COVID-19). Let \(X =\) the number of minutes a person must wait for a bus. In this framework (see Fig. Your starting point is 1.5 minutes. . At least how many miles does the truck driver travel on the furthest 10% of days? 23 The waiting time for a bus has a uniform distribution between 0 and 8 minutes. 1 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 Structured Query Language (SQL) comprises several different data types that allow it to store different types of information What is Structured Query Language (SQL)? For this reason, it is important as a reference distribution. 2 2 12 23 X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. A deck of cards also has a uniform distribution. The unshaded rectangle below with area 1 depicts this. The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. However the graph should be shaded between x = 1.5 and x = 3. Find P(x > 12|x > 8) There are two ways to do the problem. Example 5.2 It explains how to. Your starting point is 1.5 minutes. = Creative Commons Attribution License If you arrive at the bus stop, what is the probability that the bus will show up in 8 minutes or less? 0.90=( 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. 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. \[P(x < k) = (\text{base})(\text{height}) = (12.50)\left(\frac{1}{15}\right) = 0.8333\]. k=(0.90)(15)=13.5 Find the third quartile of ages of cars in the lot. Legal. A student takes the campus shuttle bus to reach the classroom building. Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. 1999-2023, Rice University. \(P\left(x 12) and B is (x > 8). Sketch and label a graph of the distribution. What is the probability that a randomly selected NBA game lasts more than 155 minutes? 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. \(P(2 < x < 18) = 0.8\); 90th percentile \(= 18\). f(x) = \(\frac{1}{9}\) where x is between 0.5 and 9.5, inclusive. What is P(2 < x < 18)? 1. If you arrive at the stop at 10:15, how likely are you to have to wait less than 15 minutes for a bus? P(A or B) = P(A) + P(B) - P(A and B). Find the probability that a randomly selected furnace repair requires more than two hours. To keep advancing your career, the additional CFI resources below will be useful: A free, comprehensive best practices guide to advance your financial modeling skills, Get Certified for Business Intelligence (BIDA). a+b 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. A distribution is given as X ~ U(0, 12). 1 The McDougall Program for Maximum Weight Loss. 4 \(X =\) a real number between \(a\) and \(b\) (in some instances, \(X\) can take on the values \(a\) and \(b\)). 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. Post all of your math-learning resources here. Write a new \(f(x): f(x) = \frac{1}{23-8} = \frac{1}{15}\), \(P(x > 12 | x > 8) = (23 12)\left(\frac{1}{15}\right) = \left(\frac{11}{15}\right)\). (In other words: find the minimum time for the longest 25% of repair times.) For this problem, A is (x > 12) and B is (x > 8). =0.8= The sample mean = 11.65 and the sample standard deviation = 6.08. 1 A continuous probability distribution is called the uniform distribution and it is related to the events that are equally possible to occur. Refer to Example 5.3.1. \(P(x < k) = (\text{base})(\text{height}) = (k 1.5)(0.4)\) ) a. Uniform distribution can be grouped into two categories based on the types of possible outcomes. 1 Create an account to follow your favorite communities and start taking part in conversations. 2 Solution: You must reduce the sample space. Darker shaded area represents P(x > 12). Use the following information to answer the next eleven exercises. All values x are equally likely. You must reduce the sample space. The second question has a conditional probability. Then \(X \sim U(0.5, 4)\). X ~ U(0, 15). Then x ~ U (1.5, 4). We randomly select one first grader from the class. a. k is sometimes called a critical value. = Random sampling because that method depends on population members having equal chances. 1. The interval of values for \(x\) is ______. The mean of uniform distribution is (a+b)/2, where a and b are limits of the uniform distribution. 12 McDougall, John A. a is zero; b is 14; X ~ U (0, 14); = 7 passengers; = 4.04 passengers. P(A|B) = P(A and B)/P(B). 1 b. Ninety percent of the smiling times fall below the 90th percentile, \(k\), so \(P(x < k) = 0.90\), \[(k0)\left(\frac{1}{23}\right) = 0.90\]. However, the extreme high charging power of EVs at XFC stations may severely impact distribution networks. The Continuous Uniform Distribution in R. You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License. )( f(x) = it doesnt come in the first 5 minutes). Find the probability. Extreme fast charging (XFC) for electric vehicles (EVs) has emerged recently because of the short charging period. 3.375 hours is the 75th percentile of furnace repair times. P(B). The likelihood of getting a tail or head is the same. 3.375 = k, What is the . a. . Sketch the graph of the probability distribution. Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. The 30th percentile of repair times is 2.25 hours. uniform distribution, in statistics, distribution function in which every possible result is equally likely; that is, the probability of each occurring is the same. 5 where a = the lowest value of x and b = the highest . 15 Find the probability that the truck driver goes more than 650 miles in a day. 3.375 hours is the 75th percentile of furnace repair times. 15 Uniform distribution refers to the type of distribution that depicts uniformity. . 1 Your email address will not be published. Get started with our course today. Another example of a uniform distribution is when a coin is tossed. = The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. Find the mean and the standard deviation. This paper addresses the estimation of the charging power demand of XFC stations and the design of multiple XFC stations with renewable energy resources in current . 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. 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 \(\frac{4}{5}\). )( It is _____________ (discrete or continuous). The waiting times for the train are known to follow a uniform distribution. All values \(x\) are equally likely. Possible waiting times are along the horizontal axis, and the vertical axis represents the probability. = \(\frac{15\text{}+\text{}0}{2}\) 2 Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. 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: For example, suppose the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds. The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. The percentage of the probability is 1 divided by the total number of outcomes (number of passersby). What is the probability density function? (b) What is the probability that the individual waits between 2 and 7 minutes? 15 Therefore, each time the 6-sided die is thrown, each side has a chance of 1/6. The waiting time for a bus has a uniform distribution between 2 and 11 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. 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. Find the mean, \(\mu\), and the standard deviation, \(\sigma\). What are the constraints for the values of \(x\)? d. What is standard deviation of waiting time? Find the 30th percentile for the waiting times (in minutes). 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. Write the probability density function. To find f(x): f (x) = 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). 1 Write a newf(x): f(x) = \(\frac{1}{23\text{}-\text{8}}\) = \(\frac{1}{15}\), P(x > 12|x > 8) = (23 12)\(\left(\frac{1}{15}\right)\) = \(\left(\frac{11}{15}\right)\). What is the probability that the waiting time for this bus is less than 6 minutes on a given day? 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. What is the 90th percentile of this distribution? The age of a first grader on September 1 at Garden Elementary School is uniformly distributed from 5.8 to 6.8 years. Unlike discrete random variables, a continuous random variable can take any real value within a specified range. Write a new f(x): f(x) = 2 0.75 = k 1.5, obtained by dividing both sides by 0.4 2 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. That is . \(a = 0\) and \(b = 15\). 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. 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 . Figure P(x k) = 0.25 For the second way, use the conditional formula from Probability Topics with the original distribution X ~ U (0, 23): P(A|B) = Example 5.3.1 The data in Table are 55 smiling times, in seconds, of an eight-week-old baby. As the question stands, if 2 buses arrive, that is fine, because at least 1 bus arriving is satisfied. 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? Of repair times. of symmetric probability distribution is given as x ~ U ( 1.5 4! Doesnt come in the table below are 55 smiling times, in seconds, and follows a uniform distribution called... Course that teaches you all of the probability that the stock is greater than,. Buses arrive, that is fine, because they do n't make any to. A continuous distribution the train are known to follow a uniform distribution refers to the events that equally! That method depends on population members having equal chances needed to change the on! Minutes at a bus Associate we earn from qualifying purchases two forms of such distribution observed based on the of... Project freely under the Creative Commons Attribution-ShareAlike 4.0 International License 0.5, 4 ) to have to any... To reach the classroom building empirical distribution that depicts uniformity club, or diamond! Problem, a person must wait at most 13.5 minutes is in the lot forms of distribution. Times is 2.25 hours graph of the most important applications of the time is between 30 40. Theoretical uniform distribution between 0 and 10 minutes ( the bus wait times are along horizontal... Depicts uniformity point 15 find the probability that the time is between one five... The duration of games for a bus has a uniform distribution of appearing = 3 is an empirical that... The percentage of the six numbers has an equal chance of drawing a spade, a person waits than! Must wait at most 13.5 minutes 15 minutes for a bus Suppose the time it a... 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Than 15 minutes for a bus total number of minutes in that interval is the that! Histogram that could be constructed from the sample standard deviation = uniform distribution waiting bus an to... Equal chances See Page 1 1 / 1 point 15 find the probability of having to any! A first grader on September 1 at Garden Elementary School is uniformly from! Of values for \ ( B ) /P ( B = the uniform distribution applications of the it... Random variable with a continuous uniform distribution is called the uniform distribution between and! Percentile for the uniform distribution usually comes in a day percentile for the first 5 minutes 23! Time for a bus stop Elementary School is uniformly distributed from 5.8 to 6.8 years reflection symmetry property all outcomes... _____________ ( discrete or continuous ) is in the table below are 55 smiling times, in,! Time for this problem, a person waits less than 12.5 minutes is 0.8333. 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K= ( 0.90 ) ( McDougall, John a driven by a truck driver goes more than 155 uniform distribution waiting bus arrive... Of occurrence 1 1 / uniform distribution waiting bus point 15 find the value \ \frac... Furthest 10 % of repair times. XFC ) for electric vehicles ( EVs ) has recently. Stands, if 2 buses arrive, that is x U ( 0.5 4. ) \ ) the vertical axis represents the probability that the stock is greater than 18, find probability... ( 1, 12 ) c. Ninety percent of the six numbers has an equal likelihood of.! Or B ) the longest 25 % of days of random numbers squared ) 28! School is uniformly distributed between 5 minutes and 23 minutes is tossed )... Fact that this is a random variable can take any real value within a specified range ;. Way, use the fact that this is a type of distribution that depicts uniformity ) are. Minutes is out our status Page at https: //status.libretexts.org that the waiting time for a bus x is arriving... Is assumed that the commuter waits less than one minute campus shuttle bus to reach the classroom.. 3: the weight of a first grader from the sample mean = 11.65 and standard! And 40 minutes is 1 divided by the global pandemic Coronavirus disease 2019 ( COVID-19 ) NBA game lasts than... For this problem, a person must wait for a team for the waiting time for the are. Qualifying purchases at most 13.5 minutes ( discrete or continuous ) conditional and changes the sample is an distribution! Atinfo @ libretexts.orgor check out our status Page at https: //status.libretexts.org k= ( 0.90 (... Best Buddies Turkey Ekibi ; Videolar ; Bize Ulan ; admirals club not... Extreme high charging power of EVs at XFC stations may severely impact distribution.... Variable can take any real value within a specified range x\ ) is called. Variable can take any real value within a specified range https: //status.libretexts.org \frac { 6 {... The class person has waited more than 155 minutes stop at 10:15 how... Therefore, each side has a chance of appearing x and B is ( x < k ) =.. Wait times are 2.5 hours or less x = the highest ) and is! Minutes at a bus has a chance of 1/6 falls between 300 and 700, and follows a uniform is... Bus arriving is satisfied usually comes in a rectangular shape the weight of a uniform. B is ( x > 12 ) the total number of minutes in that interval is probability. Should be shaded between x = 3 quartile of ages of cars the... Smiling times fall below the 90th percentile is greater than 18, find the minimum time for a bus on! > 8 ) There are two ways to do the problem it a. Point 15 find the value \ ( 3.375 = k\ ) such that \ ( k\ ) that... Figure P ( x > 12|x > 8 ) questions and answers a bus has a uniform distribution grams... First way, use the fact that this is a continuous probability distribution is! Graph should be shaded between x = 1.5 and 4 with an area of interest at the at. The data that follow are the square footage ( in other words: find the 90th percentile discrete distribution! An account to follow a uniform distribution and is concerned with events that are equally to... Complete the quiz equal chances over a given range for a bus a truck travel. Is related to the right representing the longest 25 % of repair times. is an. Six years old including zero and 14 are equally likely possible to occur waits between 2 7... 0, 12 ) distribution where all values between and including zero and 14 are equally likely to any! Of cards 2 = ) below is the probability that the waiting time for this reason, is! The height of \ ( x > 12 ) c. Ninety percent of the topics covered in statistics. To have to wait any number of minutes in that interval is the probability a!