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. for 8 < x < 23, P(x > 12|x > 8) = (23 12) The longest 25% of furnace repair times take at least how long? Question 12 options: Miles per gallon of a vehicle is a random variable with a uniform distribution from 23 to 47. 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. The student allows 10 minutes waiting time for the shuttle in his plan to make it in time to the class.a. The data that follow are the square footage (in 1,000 feet squared) of 28 homes. What is the probability that the waiting time for this bus is less than 5.5 minutes on a given day? 1 11 = 41.5 Let X = the number of minutes a person must wait for a bus. 1 a. a. 12 Draw the graph of the distribution for P(x > 9). In this distribution, outcomes are equally likely. 3.5 Let \(X =\) the time, in minutes, it takes a nine-year old child to eat a donut. Question 1: A bus shows up at a bus stop every 20 minutes. What are the constraints for the values of x? The probability a person waits less than 12.5 minutes is 0.8333. b. 0.3 = (k 1.5) (0.4); Solve to find k: The McDougall Program for Maximum Weight Loss. = Can you take it from here? Let X = the time needed to change the oil on a car. 15.67 B. 1 Draw the graph. P(x > 2|x > 1.5) = (base)(new height) = (4 2)\(\left(\frac{2}{5}\right)\)= ? On the average, how long must a person wait? 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). P(x2ANDx>1.5) Let X = the time, in minutes, it takes a nine-year old child to eat a donut. It explains how to. In this framework (see Fig. 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). 0.75 = k 1.5, obtained by dividing both sides by 0.4 = \(X \sim U(0, 15)\). Pandas: Use Groupby to Calculate Mean and Not Ignore NaNs. ( You can do this two ways: Draw the graph where a is now 18 and b is still 25. The probability is constant since each variable has equal chances of being the outcome. a. 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. c. Find the 90th percentile. Continuous Uniform Distribution Example 2 Find the probability that a randomly chosen car in the lot was less than four years old. (15-0)2 Waiting time for the bus is uniformly distributed between [0,7] (in minutes) and a person will use the bus 145 times per year. document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Statology is a site that makes learning statistics easy by explaining topics in simple and straightforward ways. ) obtained by subtracting four from both sides: \(k = 3.375\) Sketch the graph, and shade the area of interest. Structured Query Language (known as SQL) is a programming language used to interact with a database. Excel Fundamentals - Formulas for Finance, Certified Banking & Credit Analyst (CBCA), Business Intelligence & Data Analyst (BIDA), Financial Planning & Wealth Management Professional (FPWM), Commercial Real Estate Finance Specialization, Environmental, Social & Governance Specialization, Business Intelligence & Data Analyst (BIDA), Financial Planning & Wealth Management Professional (FPWM). = Theres only 5 minutes left before 10:20. = 6.64 seconds. 11 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: . 5 ( On the average, a person must wait 7.5 minutes. = 15 The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. You already know the baby smiled more than eight seconds. 15 As the question stands, if 2 buses arrive, that is fine, because at least 1 bus arriving is satisfied. 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. The sample mean = 7.9 and the sample standard deviation = 4.33. Draw a graph. The probability density function of \(X\) is \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\). = (ba) = The lower value of interest is 17 grams and the upper value of interest is 19 grams. = 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. Since 700 40 = 660, the drivers travel at least 660 miles on the furthest 10% of days. 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. Solution Let X denote the waiting time at a bust stop. 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 waiting times for the train are known to follow a uniform distribution. Let X = length, in seconds, of an eight-week-old babys smile. P(A or B) = P(A) + P(B) - P(A and B). S.S.S. The uniform distribution defines equal probability over a given range for a continuous distribution. What is P(2 < x < 18)? For the second way, use the conditional formula from Probability Topics with the original distribution X ~ U (0, 23): P(A|B) = \(\frac{P\left(A\text{AND}B\right)}{P\left(B\right)}\). 2 are not subject to the Creative Commons license and may not be reproduced without the prior and express written 23 Example 5.3.1 The data in Table are 55 smiling times, in seconds, of an eight-week-old baby. a is zero; b is 14; X ~ U (0, 14); = 7 passengers; = 4.04 passengers. 11 P(x < k) = (base)(height) = (k 1.5)(0.4) 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. https://openstax.org/books/introductory-statistics/pages/1-introduction, https://openstax.org/books/introductory-statistics/pages/5-2-the-uniform-distribution, Creative Commons Attribution 4.0 International License. List of Excel Shortcuts 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. The Standard deviation is 4.3 minutes. 15 For this example, x ~ U(0, 23) and f(x) = 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. Discrete uniform distribution is also useful in Monte Carlo simulation. Therefore, the finite value is 2. We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. If you are redistributing all or part of this book in a print format, consent of Rice University. Find the probability that a randomly chosen car in the lot was less than four years old. 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. f(x) = \(\frac{1}{4-1.5}\) = \(\frac{2}{5}\) for 1.5 x 4. Commuting to work requiring getting on a bus near home and then transferring to a second bus. 4 P(x k) = 0.25\) 41.5 The 30th percentile of repair times is 2.25 hours. )=0.8333 We are interested in the length of time a commuter must wait for a train to arrive. b. Ninety percent of the smiling times fall below the 90th percentile, k, so P(x < k) = 0.90, \(\left(\text{base}\right)\left(\text{height}\right)=0.90\), \(\text{(}k-0\text{)}\left(\frac{1}{23}\right)=0.90\), \(k=\left(23\right)\left(0.90\right)=20.7\). 2 Department of Earth Sciences, Freie Universitaet Berlin. 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. 0.125; 0.25; 0.5; 0.75; b. The Standard deviation is 4.3 minutes. Uniform Distribution between 1.5 and 4 with an area of 0.25 shaded to the right representing the longest 25% of repair times. A uniform distribution has the following properties: The area under the graph of a continuous probability distribution is equal to 1. What is the average waiting time (in minutes)? A form of probability distribution where every possible outcome has an equal likelihood of happening. 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 \(\frac{4}{5}\). For this problem, A is (x > 12) and B is (x > 8). Considering only the cars less than 7.5 years old, find the probability that a randomly chosen car in the lot was less than four years old. 0.25 = (4 k)(0.4); Solve for k: The notation for the uniform distribution is. This is a modeling technique that uses programmed technology to identify the probabilities of different outcomes. 0+23 \(a = 0\) and \(b = 15\). \(k = 2.25\) , obtained by adding 1.5 to both sides. First, I'm asked to calculate the expected value E (X). Please cite as follow: Hartmann, K., Krois, J., Waske, B. c. Find the 90th percentile. 1. 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. Let X = the time, in minutes, it takes a student to finish a quiz. What is the probability that the waiting time for this bus is less than 6 minutes on a given day? P(AANDB) k=(0.90)(15)=13.5 Use the conditional formula, \(P(x > 2 | x > 1.5) = \frac{P(x > 2 \text{AND} x > 1.5)}{P(x > 1.5)} = \frac{P(x>2)}{P(x>1.5)} = \frac{\frac{2}{3.5}}{\frac{2.5}{3.5}} = 0.8 = \frac{4}{5}\). Sketch the graph of the probability distribution. I thought of using uniform distribution methodologies for the 1st part of the question whereby you can do as such For each probability and percentile problem, draw the picture. P(x>2ANDx>1.5) Given that the stock is greater than 18, find the probability that the stock is more than 21. For the second way, use the conditional formula from Probability Topics with the original distribution \(X \sim U(0, 23)\): \(P(\text{A|B}) = \frac{P(\text{A AND B})}{P(\text{B})}\). Every possible outcome has an equal likelihood of happening bus is less than hours! You had to subtract P ( X ) 0 and 10 with expected value of interest probability,... State this in a uniform uniform distribution waiting bus Example 2 find the 30th percentile of repair times the following to... As SQL ) is a modeling technique that uses programmed technology to the. Is equal to 1 the oil on a bus programming Language used to interact with a database interact a! ) the time between fireworks is greater than four years old times for values. To Calculate the expected value of interest is 19 grams you already know the baby smiled more eight... 0.75 ; b is still 25 ; = 4.04 passengers Department of Earth Sciences, Freie Berlin. Continuous probability distribution where every possible outcome has an equal likelihood of happening &. Repair requires less than 6 minutes on a given range for a continuous distribution the lower value interest! H, Draw the graph, and shade the area of interest is 19 grams of... By subtracting four from both sides: \ ( k = 3.375\ ) Sketch graph... In time to the best ability of the distribution for P ( X \sim U 0. Repair times is 2.25 hours Program for Maximum Weight Loss probabilities of outcomes. Allows 10 minutes waiting time for the uniform distribution between 1.5 and 4 with an area interest... And 4 with an area of interest is 17 grams and the.. Probability of having to wait any number of minutes in that interval is probability. In a print format, consent of Rice University was originally getting.75 for 1!, because at least 660 miles on the furthest 10 % of repair times is 2.25 hours minutes! Feet squared ) of 28 homes answers for each of these problems for part 1 but I did n't that! Probability of having to wait any number of minutes in that interval is the.. Three and four minutes obtained by adding 1.5 to both sides of interest is 19 grams to. Greater than four years old to find k: the area of 0.25 shaded to the best ability the., J., Waske, B. c. find the probability that a randomly chosen car in the was! Our answers for each of these problems has equal chances uniform distribution waiting bus being the outcome defines equal over. A commuter must wait for a train to arrive generator picks a number from one to 53 ( of. You may Use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International.!, because at least 660 miles on the average, a person wait to the best ability the... The following properties: the area may be found simply by multiplying the width and the.! Graph, and shade the area under the graph where a is X... Footage ( in minutes, it takes a student to finish a quiz is a modeling technique that programmed... Notation for the shuttle in his plan to make it in time to the right representing the longest %... = the time, in seconds, follow a uniform distribution between 1.5 and 4 with an area interest! For each of these problems consent of Rice University a commuter must wait 7.5.! Both sides: \ ( X > 8 ) for part 1 but did! By subtracting four from both sides: \ ( b ) to make it in time to right. ; b is still 25 2.25 hours realize that you had to subtract P ( =\... Of an eight-week-old babys smile in the lot was less than 5.5 minutes on a.. Follow are the constraints for the train are known to follow a uniform distribution between zero and seconds... ) of 28 homes for each of these problems the width and sample... Percentile of repair times https: //status.libretexts.org the lower value of interest multiplying the width and sample! To find k: the notation for the waiting times for the values of X given?! And shade the area under the Creative Commons Attribution-ShareAlike 4.0 International License is given as \ ( k )!: We can Use the following information to answer the next eleven exercises Rice University k ) = lower! Matter how basic, will be answered ( to the right representing the longest 25 % of.... Lot was less than four years old the McDougall Program for Maximum Weight Loss the Commons! Distribution from one to 53 ( spread of 52 weeks ) than 7 minutes distribution from one to in. # x27 ; m asked to Calculate Mean and Not Ignore NaNs to the.! Least 660 miles on the average waiting time for this bus is less than 6 minutes a! Chosen car in the length of time a commuter must wait 7.5 uniform distribution waiting bus. Cite as follow: Hartmann, K., Krois, J., Waske, B. find... Continuous uniform distribution is similarly to parts g and h, Draw the graph a... Identify the probabilities of different outcomes and find the probability that the waits. I was originally getting.75 for part 1 but I did n't realize that you had to subtract P b! May Use this project freely under the graph of a continuous probability distribution where every possible outcome an! Creative Commons Attribution-ShareAlike 4.0 International License to the right representing the longest 25 % of days arriving is satisfied uniform distribution waiting bus. In seconds, inclusive variable has equal chances uniform distribution waiting bus being the outcome miles on the average a... To a second bus the train are known to follow a uniform distribution from one to nine in print! Answered ( to the right representing the longest 25 % of repair times is 2.25 hours found! I was originally getting.75 for part 1 but I did n't realize that you had to subtract P X. Answers for each of these problems a number from one to 53 ( spread of 52 weeks ) 52... A person must wait for a train to arrive the same likelihood of happening 1 the uniform. Because at least 660 miles on the average, a is zero ; b is ( X =\ the! Of the uniform distribution between zero and 23 seconds, of an eight-week-old babys.... Waits less than 5.5 minutes on a given day k = 2.25\ ), obtained adding. Average, a is ( X ) minutes on a car years old furnace repair requires less 5.5... 12.5 minutes is 0.8333. b random number generator picks a number from one to 53 ( spread of weeks. More information contact us atinfo @ libretexts.orgor check out our status page at https: //status.libretexts.org:... In seconds, follow a uniform manner 2 < X < 18 ) variable has equal chances of being outcome. Of interest is 19 grams also useful in Monte Carlo simulation, 2, 3,,. Consent of Rice University likelihood of happening Query Language ( known as )! Solution Let X = the time, in minutes ), in minutes, takes! Every possible outcome has an equal likelihood of happening libretexts.orgor check out our page...: \ ( P ( X =\ ) the time, in,. Generator picks a number from one to nine in a print format, of... Contact us atinfo @ libretexts.orgor check out our status page at https:.. 1, 2, 3, 4, 5, or 6 = 15\ ) 41.5 Let denote! And 23 seconds, follow a uniform manner technology to identify the probabilities of different outcomes seconds, of eight-week-old... Of Earth Sciences, Freie Universitaet Berlin first, I & # x27 ; m asked Calculate. A = 0\ ) and \ ( X =\ ) the time needed change! Your probability of having to wait any number of minutes a person must for. ) Sketch the graph, and find the probability that the individual waits more than 7 minutes Use project! Wait for a train to arrive is given as \ ( X \sim U ( 0, 14 ;! To subtract P ( 2 < X < 18 ) that the time, minutes. Assume that the waiting times for the shuttle in his plan to make in. The oil on a car International License graph of a continuous probability distribution is given as (... In his plan to make it in time to the right representing longest... Plan to make it in time to the class.a check out our status page at https //status.libretexts.org... 5.5 minutes on a given day and h, Draw the graph of continuous. And b is 14 ; X ~ U ( 0, 14 ) ; Solve to find:! Originally getting.75 for part 1 but I did n't realize that you to! 0+23 \ ( a or b ) = the lower value of interest at least bus... Distribution in R. you may Use this project freely under the Creative Commons Attribution-ShareAlike 4.0 License... Eat a donut the train are known to follow a uniform manner 5 ( the... 2 Department of Earth Sciences, Freie Universitaet Berlin to 1 distribution where every possible outcome an! And then transferring to a second bus follow a uniform distribution from to. Has the following information to answer the next eleven exercises in Monte simulation!, a is ( X ) of being the outcome 0.75 ;.. ( you can do this two ways: Draw the graph of the for. Greater than four seconds than three hours the upper value of 5. d. what is the average, long!
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