= 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. 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. 3 buses will arrive at the the same time (i.e. { "5.01:_Introduction" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "5.02:_Continuous_Probability_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "5.03:_The_Uniform_Distribution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "5.04:_The_Exponential_Distribution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "5.05:_Continuous_Distribution_(Worksheet)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "5.E:_Continuous_Random_Variables_(Exercises)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "5.E:_Exercises" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Sampling_and_Data" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Descriptive_Statistics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Probability_Topics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Discrete_Random_Variables" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Continuous_Random_Variables" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_The_Normal_Distribution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_The_Central_Limit_Theorem" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Confidence_Intervals" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Hypothesis_Testing_with_One_Sample" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Hypothesis_Testing_with_Two_Samples" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11:_The_Chi-Square_Distribution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12:_Linear_Regression_and_Correlation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "13:_F_Distribution_and_One-Way_ANOVA" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "authorname:openstax", "showtoc:no", "license:ccby", "Uniform distribution", "program:openstax", "licenseversion:40", "source@https://openstax.org/details/books/introductory-statistics" ], https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FIntroductory_Statistics%2FBook%253A_Introductory_Statistics_(OpenStax)%2F05%253A_Continuous_Random_Variables%2F5.03%253A_The_Uniform_Distribution, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), source@https://openstax.org/details/books/introductory-statistics, status page at https://status.libretexts.org. Public transport systems have been affected by the global pandemic Coronavirus disease 2019 (COVID-19). X ~ U(0, 15). f(x) = We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. P(x>12ANDx>8) a. 11 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. Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. . Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. The sample mean = 2.50 and the sample standard deviation = 0.8302. 12= We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. Ninety percent of the time, a person must wait at most 13.5 minutes. For this example, x ~ U(0, 23) and f(x) = 1 The height is \(\frac{1}{\left(25-18\right)}\) = \(\frac{1}{7}\). = Let x = the time needed to fix a furnace. For example, it can arise in inventory management in the study of the frequency of inventory sales. Sixty percent of commuters wait more than how long for the train? Find the probability. The cumulative distribution function of X is P(X x) = \(\frac{x-a}{b-a}\). Use the following information to answer the next eleven exercises. Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. = f(x) = The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. 3.5 In this distribution, outcomes are equally likely. The possible values would be 1, 2, 3, 4, 5, or 6. X is continuous. b. It is defined by two different parameters, x and y, where x = the minimum value and y = the maximum value. Write the answer in a probability statement. Draw the graph. 15 Darker shaded area represents P(x > 12). Let k = the 90th percentile. A random number generator picks a number from one to nine in a uniform manner. (b) The probability that the rider waits 8 minutes or less. 2 The lower value of interest is 155 minutes and the upper value of interest is 170 minutes. Find the probability that a randomly selected furnace repair requires less than three hours. What are the constraints for the values of \(x\)? 15 \nonumber\]. Solution 2: The minimum time is 120 minutes and the maximum time is 170 minutes. What is the probability that the waiting time for this bus is less than 6 minutes on a given day? The longest 25% of furnace repair times take at least how long? = Required fields are marked *. obtained by subtracting four from both sides: k = 3.375. ) 15 Then \(X \sim U(0.5, 4)\). Find the 90th percentile for an eight-week-old baby's smiling time. The probability density function is The notation for the uniform distribution is. Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. 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. c. Find the 90th percentile. 12 = (230) 23 the 1st and 3rd buses will arrive in the same 5-minute period)? At least how many miles does the truck driver travel on the furthest 10% of days? The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer). = 7.5. 2 What is the height of f(x) for the continuous probability distribution? The area must be 0.25, and 0.25 = (width)\(\left(\frac{1}{9}\right)\), so width = (0.25)(9) = 2.25. One of the most important applications of the uniform distribution is in the generation of random numbers. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. Write a new f(x): f(x) = State the values of a and b. 1 The concept of uniform distribution, as well as the random variables it describes, form the foundation of statistical analysis and probability theory. P(x>2ANDx>1.5) The possible outcomes in such a scenario can only be two. It is because an individual has an equal chance of drawing a spade, a heart, a club, or a diamond. 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. 12 When working out problems that have a uniform distribution, be careful to note if the data are inclusive or exclusive of endpoints. 12 The waiting time for a bus has a uniform distribution between 0 and 10 minutes. Find the 90th percentile. 1 A form of probability distribution where every possible outcome has an equal likelihood of happening. 2 I thought of using uniform distribution methodologies for the 1st part of the question whereby you can do as such \(P(x > k) = (\text{base})(\text{height}) = (4 k)(0.4)\) What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds? c. Ninety percent of the time, the time a person must wait falls below what value? 1.0/ 1.0 Points. Find the 90th percentile. 15 2 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. 1 a. For the second way, use the conditional formula from Probability Topics with the original distribution X ~ U (0, 23): P(A|B) = \(P\left(x 2|x > 1.5) = (\text{base})(\text{new height}) = (4 2)(25)\left(\frac{2}{5}\right) =\) ? A student takes the campus shuttle bus to reach the classroom building. Find the probability. obtained by subtracting four from both sides: k = 3.375 Questions, no matter how basic, will be answered (to the best ability of the online subscribers). On the average, how long must a person wait? Example 1 The data in the table below are 55 smiling times, in seconds, of an eight-week-old baby. = The Continuous Uniform Distribution in R. You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License. b. Ninety percent of the smiling times fall below the 90th percentile, k, so P(x < k) = 0.90. P(A|B) = P(A and B)/P(B). In this framework (see Fig. \(k\) is sometimes called a critical value. Then X ~ U (6, 15). The uniform distribution is a probability distribution in which every value between an interval from a to b is equally likely to occur. The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution. ) 23 For this example, \(X \sim U(0, 23)\) and \(f(x) = \frac{1}{23-0}\) for \(0 \leq X \leq 23\). 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 Find the probability that a randomly chosen car in the lot was less than four years old. = The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. Learn more about how Pressbooks supports open publishing practices. The total duration of baseball games in the major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive. We are interested in the weight loss of a randomly selected individual following the program for one month. ba Sketch and label a graph of the distribution. obtained by dividing both sides by 0.4 You already know the baby smiled more than eight seconds. 1 0.625 = 4 k, 2 f ( x) = 1 12 1, 1 x 12 = 1 11, 1 x 12 = 0.0909, 1 x 12. Entire shaded area shows P(x > 8). Use the conditional formula, P(x > 2|x > 1.5) = \(\frac{P\left(x>2\text{AND}x>1.5\right)}{P\left(x>\text{1}\text{.5}\right)}=\frac{P\left(x>2\right)}{P\left(x>1.5\right)}=\frac{\frac{2}{3.5}}{\frac{2.5}{3.5}}=\text{0}\text{.8}=\frac{4}{5}\). Then X ~ U (0.5, 4). 2 P(2 < x < 18) = (base)(height) = (18 2) However, if you favored short people or women, they would have a higher chance of being given the $100 bill than the other passersby. Solve the problem two different ways (see Example 5.3). Thought I would just take the integral of 1/60 dx from 15 to 30, but is. 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