(a) The probability density function of X is. = a is zero; b is 14; X ~ U (0, 14); = 7 passengers; = 4.04 passengers. Find the probability that the truck driver goes more than 650 miles in a day. \(X =\) __________________. 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. a person has waited more than four minutes is? The standard deviation of X is \(\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}\). 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. = Post all of your math-learning resources here. Note: We can use the Uniform Distribution Calculator to check our answers for each of these problems. b. admirals club military not in uniform. In their calculations of the optimal strategy . = Find the mean, , and the standard deviation, . b. = \(\frac{a\text{}+\text{}b}{2}\) What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? 1 Let \(x =\) the time needed to fix a furnace. . Sketch the graph, and shade the area of interest. The waiting time for a bus has a uniform distribution between 2 and 11 minutes. 41.5 What is the variance?b. 1 = Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. The waiting times for the train are known to follow a uniform distribution. Another simple example is the probability distribution of a coin being flipped. b. . The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. A continuous uniform distribution (also referred to as rectangular distribution) is a statistical distribution with an infinite number of equally likely measurable values. 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 . 233K views 3 years ago This statistics video provides a basic introduction into continuous probability distribution with a focus on solving uniform distribution problems. = 7.5. The probability P(c < X < d) may be found by computing the area under f(x), between c and d. Since the corresponding area is a rectangle, the area may be found simply by multiplying the width and the height. Figure Find the 90th percentile for an eight-week-old baby's smiling time. 1 1999-2023, Rice University. What does this mean? 150 Then X ~ U (6, 15). \(f\left(x\right)=\frac{1}{8}\) where \(1\le x\le 9\). \(P\left(x 9). What is the theoretical standard deviation? What are the constraints for the values of \(x\)? 230 \(f(x) = \frac{1}{4-1.5} = \frac{2}{5}\) for \(1.5 \leq x \leq 4\). 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 . You already know the baby smiled more than eight seconds. 2.75 The graph illustrates the new sample space. Find the probability. \(X =\) a real number between \(a\) and \(b\) (in some instances, \(X\) can take on the values \(a\) and \(b\)). P(AANDB) b is 12, and it represents the highest value of x. The longest 25% of furnace repair times take at least how long? k is sometimes called a critical value. The data that follow are the number of passengers on 35 different charter fishing boats. 15 b. Ninety percent of the smiling times fall below the 90th percentile, \(k\), so \(P(x < k) = 0.90\), \[(k0)\left(\frac{1}{23}\right) = 0.90\]. Get started with our course today. 5. 15 In order for a bus to come in the next 15 minutes, that means that it has to come in the last 5 minutes of 10:00-10:20 OR it has to come in the first 10 minutes of 10:20-10:40. What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds? X = a real number between a and b (in some instances, X can take on the values a and b). The student allows 10 minutes waiting time for the shuttle in his plan to make it in time to the class.a. P(A or B) = P(A) + P(B) - P(A and B). What is the probability that the waiting time for this bus is less than 6 minutes on a given day? Waiting time for the bus is uniformly distributed between [0,7] (in minutes) and a person will use the bus 145 times per year. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. (15-0)2 So, \(P(x > 12|x > 8) = \frac{(x > 12 \text{ AND } x > 8)}{P(x > 8)} = \frac{P(x > 12)}{P(x > 8)} = \frac{\frac{11}{23}}{\frac{15}{23}} = \frac{11}{15}\). Example The data in the table below are 55 smiling times, in seconds, of an eight-week-old baby. Find the probability that the value of the stock is between 19 and 22. This is a modeling technique that uses programmed technology to identify the probabilities of different outcomes. Continuous Uniform Distribution Example 2 For the first way, use the fact that this is a conditional and changes the sample space. You must reduce the sample space. Write the probability density function. (k0)( 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 density function of X is \(f\left(x\right)=\frac{1}{b-a}\) for a x b. 1 In this distribution, outcomes are equally likely. It would not be described as uniform probability. It means every possible outcome for a cause, action, or event has equal chances of occurrence. 15 \(X\) is continuous. 2 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? Let x = the time needed to fix a furnace. Let \(X =\) the time needed to change the oil in a car. Uniform distribution has probability density distributed uniformly over its defined interval. 2 Find the probability that the time is more than 40 minutes given (or knowing that) it is at least 30 minutes. Second way: Draw the original graph for X ~ U (0.5, 4). Find the mean, , and the standard deviation, . a+b All values \(x\) are equally likely. ) Continuous Uniform Distribution - Waiting at the bus stop 1,128 views Aug 9, 2020 20 Dislike Share The A Plus Project 331 subscribers This is an example of a problem that can be solved with the. The data in [link] are 55 smiling times, in seconds, of an eight-week-old baby. For this problem, \(\text{A}\) is (\(x > 12\)) and \(\text{B}\) is (\(x > 8\)). The sample mean = 11.65 and the sample standard deviation = 6.08. Answer: a. \(k\) is sometimes called a critical value. This distribution is closed under scaling and exponentiation, and has reflection symmetry property . a. The 90th percentile is 13.5 minutes. c. What is the expected waiting time? 23 11 On the average, a person must wait 7.5 minutes. P(x 19) = (25 19) \(\left(\frac{1}{9}\right)\) 1. 1 Example 5.2 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. b. Find the value \(k\) such that \(P(x < k) = 0.75\). = That is X U ( 1, 12). (230) Can you take it from here? Find the probability that the value of the stock is more than 19. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. . Commuting to work requiring getting on a bus near home and then transferring to a second bus. P(x>8) The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. The possible values would be 1, 2, 3, 4, 5, or 6. Your starting point is 1.5 minutes. 1 Refer to Example 5.3.1. The waiting time for a bus has a uniform distribution between 0 and 10 minutes. a+b 12 \nonumber\]. 0+23 If the waiting time (in minutes) at each stop has a uniform distribution with A = 0 and B = 5, then it can be shown that the total waiting time Y has the pdf f(y) = 1 25 y 0 y < 5 2 5 1 25 y 5 y 10 0 y < 0 or y > 10 How likely is it that a bus will arrive in the next 5 minutes? P(x>2) (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. 2 = What is the expected waiting time? The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. State the values of a and \(b\). (a) The solution is The Continuous Uniform Distribution in R. You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License. Sketch and label a graph of the distribution. (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.) ) We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. The area must be 0.25, and 0.25 = (width)\(\left(\frac{1}{9}\right)\), so width = (0.25)(9) = 2.25. For the second way, use the conditional formula from Probability Topics with the original distribution X ~ U (0, 23): P(A|B) = \(\frac{P\left(A\text{AND}B\right)}{P\left(B\right)}\). f (x) = \(\frac{1}{15\text{}-\text{}0}\) = \(\frac{1}{15}\) ( Sixty percent of commuters wait more than how long for the train? ba The second question has a conditional probability. For this example, X ~ U(0, 23) and f(x) = \(\frac{1}{23-0}\) for 0 X 23. 41.5 Let k = the 90th percentile. a. This means that any smiling time from zero to and including 23 seconds is equally likely. We write X U(a, b). )=0.90, k=( a. \(0.25 = (4 k)(0.4)\); Solve for \(k\): A student takes the campus shuttle bus to reach the classroom building. c. This probability question is a conditional. What is the 90th percentile of square footage for homes? Then x ~ U (1.5, 4). a. 1 Pandas: Use Groupby to Calculate Mean and Not Ignore NaNs. 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), 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 . Ninety percent of the time, a person must wait at most 13.5 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. 0.10 = \(\frac{\text{width}}{\text{700}-\text{300}}\), so width = 400(0.10) = 40. hours and Find the third quartile of ages of cars in the lot. Discrete uniform distributions have a finite number of outcomes. 15 Here we introduce the concepts, assumptions, and notations related to the congestion model. 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. = 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. The longest 25% of furnace repair times take at least how long? )=20.7 obtained by subtracting four from both sides: \(k = 3.375\) a. The number of values is finite. 15 Example 1 The data in the table below are 55 smiling times, in seconds, of an eight-week-old baby. A distribution is given as \(X \sim U(0, 20)\). Buses run every 30 minutes without fail, hence the next bus will come any time during the next 30 minutes with evenly distributed probability (a uniform distribution). 230 = View full document See Page 1 1 / 1 point First, I'm asked to calculate the expected value E (X). k=(0.90)(15)=13.5 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. Different outcomes a finite number of passengers on 35 different charter fishing boats of baseball games in the below! That the value \ ( X =\ ) the time, a person must wait 7.5 minutes ) where (. 0.5 and 4 minutes, inclusive graph, and the standard deviation 6.08. Between a and b ( in some instances, X can take the... 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