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. 2 Is this because of the multiple intervals (10-10:20, 10:20-10:40, etc)? The Sky Train from the terminal to the rentalcar and longterm parking center is supposed to arrive every eight 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. First, I'm asked to calculate the expected value E (X). What is the 90th percentile of square footage for homes? . The Standard deviation is 4.3 minutes. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. The cumulative distribution function of \(X\) is \(P(X \leq x) = \frac{x-a}{b-a}\). For example, in our previous example we said the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds. = 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. 1 Extreme fast charging (XFC) for electric vehicles (EVs) has emerged recently because of the short charging period. Use the conditional formula, P(x > 2|x > 1.5) = \(\frac{P\left(x>2\text{AND}x>1.5\right)}{P\left(x>\text{1}\text{.5}\right)}=\frac{P\left(x>2\right)}{P\left(x>1.5\right)}=\frac{\frac{2}{3.5}}{\frac{2.5}{3.5}}=\text{0}\text{.8}=\frac{4}{5}\). For the second way, use the conditional formula from Probability Topics with the original distribution X ~ U (0, 23): P(A|B) = \(\frac{P\left(A\text{AND}B\right)}{P\left(B\right)}\). Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. McDougall, John A. If so, what if I had wait less than 30 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: You must reduce the sample space. \(f\left(x\right)=\frac{1}{8}\) where \(1\le x\le 9\). However the graph should be shaded between \(x = 1.5\) and \(x = 3\). Statistics and Probability questions and answers A bus arrives every 10 minutes at a bus stop. 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. 150 Find the probability that the commuter waits less than one minute. Find the probability that the value of the stock is between 19 and 22. X is continuous. Write the probability density function. Let X = the time needed to change the oil on a car. Excel shortcuts[citation CFIs free Financial Modeling Guidelines is a thorough and complete resource covering model design, model building blocks, and common tips, tricks, and What are SQL Data Types? 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 11 2 Questions, no matter how basic, will be answered (to the best ability of the online subscribers). obtained by subtracting four from both sides: k = 3.375 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. ) Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. 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. (230) For the second way, use the conditional formula from Probability Topics with the original distribution \(X \sim U(0, 23)\): \(P(\text{A|B}) = \frac{P(\text{A AND B})}{P(\text{B})}\). 1.5+4 The graph of a uniform distribution is usually flat, whereby the sides and top are parallel to the x- and y-axes. The sample mean = 11.65 and the sample standard deviation = 6.08. 2 a+b Find P(X<12:5). 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 . The probability density function of \(X\) is \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\). Find the third quartile of ages of cars in the lot. ) The notation for the uniform distribution is. Find \(P(x > 12 | x > 8)\) There are two ways to do the problem. 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. However the graph should be shaded between x = 1.5 and x = 3. 2.75 The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. obtained by subtracting four from both sides: k = 3.375. For this example, \(X \sim U(0, 23)\) and \(f(x) = \frac{1}{23-0}\) for \(0 \leq X \leq 23\). for 0 x 15. 11 f (x) = The probability density function is P(x>12ANDx>8) Example The data in the table below are 55 smiling times, in seconds, of an eight-week-old baby. 11 For each probability and percentile problem, draw the picture. ) ) 23 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. What is the probability density function? 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. Standard deviation is (a-b)^2/12 = (0-12)^2/12 = (-12^2)/12 = 144/12 = 12 c. Prob (Wait for more than 5 min) = (12-5)/ (12-0) = 7/12 = 0.5833 d. Solution 1: The minimum amount of time youd have to wait is 0 minutes and the maximum amount is 20 minutes. In this distribution, outcomes are equally likely. and The waiting time at a bus stop is uniformly distributed between 1 and 12 minute. 41.5 12 a. Discrete uniform distribution is also useful in Monte Carlo simulation. One of the most important applications of the uniform distribution is in the generation of random numbers. = Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. Sketch and label a graph of the distribution. 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. X = a real number between a and b (in some instances, X can take on the values a and b). This is a conditional probability question. ( Write the probability density function. \(f(x) = \frac{1}{9}\) where \(x\) is between 0.5 and 9.5, inclusive. a= 0 and b= 15. For example, we want to predict the following: The amount of timeuntilthe customer finishes browsing and actually purchases something in your store (success). The lower value of interest is 155 minutes and the upper value of interest is 170 minutes. c. What is the expected waiting time? \nonumber\]. 15 Then X ~ U (6, 15). 3.375 hours is the 75th percentile of furnace repair times. Find the probability that a randomly selected furnace repair requires less than three hours. = Want to create or adapt books like this? The graph illustrates the new sample space. XU(0;15). \(P\left(x 2) = (base)(height) = (4 2)(0.4) = 0.8, b. P(x < 3) = (base)(height) = (3 1.5)(0.4) = 0.6. 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. (d) The variance of waiting time is . for 0 X 23. Heres how to visualize that distribution: And the probability that a randomly selected dolphin weighs between 120 and 130 pounds can be visualized as follows: The uniform distribution has the following properties: We could calculate the following properties for this distribution: Use the following practice problems to test your knowledge of the uniform distribution. 1 The Continuous Uniform Distribution in R. You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License. (a) What is the probability that the individual waits more than 7 minutes? Let k = the 90th percentile. 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 . )( Thank you! What is the 90th percentile of this distribution? a. In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. To predict the amount of waiting time until the next event (i.e., success, failure, arrival, etc.). = \(\frac{15\text{}+\text{}0}{2}\) 5 a. Creative Commons Attribution License The cumulative distribution function of X is P(X x) = \(\frac{x-a}{b-a}\). 23 The data in [link] are 55 smiling times, in seconds, of an eight-week-old baby. This may have affected the waiting passenger distribution on BRT platform space. If you randomly select a frog, what is the probability that the frog weighs between 17 and 19 grams? Financial Modeling & Valuation Analyst (FMVA), Commercial Banking & Credit Analyst (CBCA), Capital Markets & Securities Analyst (CMSA), Certified Business Intelligence & Data Analyst (BIDA), Financial Planning & Wealth Management (FPWM). 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. = 6.64 seconds. The graph of the rectangle showing the entire distribution would remain the same. You already know the baby smiled more than eight seconds. 4 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. = \(\frac{0\text{}+\text{}23}{2}\) To find f(x): f (x) = Find the probability that the time is between 30 and 40 minutes. \[P(x < k) = (\text{base})(\text{height}) = (12.50)\left(\frac{1}{15}\right) = 0.8333\]. 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. The probability is constant since each variable has equal chances of being the outcome. ) a. \(b\) is \(12\), and it represents the highest value of \(x\). The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. The 90th percentile is 13.5 minutes. What is \(P(2 < x < 18)\)? The probability of drawing any card from a deck of cards. What is P(2 < x < 18)? The waiting time at a bus stop is uniformly distributed between 1 and 12 minute. Press question mark to learn the rest of the keyboard shortcuts. a. = b. Ninety percent of the smiling times fall below the 90th percentile, k, so P(x < k) = 0.90, \(\left(\text{base}\right)\left(\text{height}\right)=0.90\), \(\text{(}k-0\text{)}\left(\frac{1}{23}\right)=0.90\), \(k=\left(23\right)\left(0.90\right)=20.7\). I'd love to hear an explanation for these answers when you get one, because they don't make any sense to me. The mean of uniform distribution is (a+b)/2, where a and b are limits of the uniform distribution. k=(0.90)(15)=13.5 Beta distribution is a well-known and widely used distribution for modeling and analyzing lifetime data, due to its interesting characteristics. 5 Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. 2 Not all uniform distributions are discrete; some are continuous. Uniform Distribution between 1.5 and 4 with an area of 0.30 shaded to the left, representing the shortest 30% of repair times. \(k = 2.25\) , obtained by adding 1.5 to both sides. Legal. 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. Uniform distribution has probability density distributed uniformly over its defined interval. 2 it doesnt come in the first 5 minutes). 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. Plume, 1995. Second way: Draw the original graph for X ~ U (0.5, 4). 15 The Uniform Distribution by OpenStaxCollege is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted. P(x>1.5) 1 . \(\mu = \frac{a+b}{2} = \frac{15+0}{2} = 7.5\). 16 A distribution is given as X ~ U (0, 20). looks like this: f (x) 1 b-a X a b. 2.1.Multimodal generalized bathtub. 2 Find the third quartile of ages of cars in the lot. 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. In this paper, a six parameters beta distribution is introduced as a generalization of the two (standard) and the four parameters beta distributions. 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. 3.375 = k, 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)=\left\{\begin{array}{cc} \frac . P(2 < x < 18) = 0.8; 90th percentile = 18. Refer to Example 5.2. For the first way, use the fact that this is a conditional and changes the sample space. P(x \(x\)) = (b x)\(\left(\frac{1}{b-a}\right)\), Area Between \(c\) and \(d\): \(P(c < x < d) = (\text{base})(\text{height}) = (d c)\left(\frac{1}{b-a}\right)\), Uniform: \(X \sim U(a, b)\) where \(a < x < b\). Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. = However the graph should be shaded between x = 1.5 and x = 3. At least how many miles does the truck driver travel on the furthest 10% of days? 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. 15 The percentage of the probability is 1 divided by the total number of outcomes (number of passersby). for a x b. 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. You already know the baby smiled more than eight seconds. What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? a+b What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds? The age of a first grader on September 1 at Garden Elementary School is uniformly distributed from 5.8 to 6.8 years. 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. b. pdf: \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\), standard deviation \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}}\), \(P(c < X < d) = (d c)\left(\frac{1}{b-a}\right)\). P(AANDB) e. \(\mu =\frac{a+b}{2}\) and \(\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}\), \(\mu =\frac{1.5+4}{2}=2.75\) The probability of waiting more than seven minutes given a person has waited more than four minutes is? Answer: (Round to two decimal place.) 12 What is the probability density function? 23 As the question stands, if 2 buses arrive, that is fine, because at least 1 bus arriving is satisfied. A continuous random variable X has a uniform distribution, denoted U ( a, b), if its probability density function is: f ( x) = 1 b a. for two constants a and b, such that a < x < b. 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)? P(B) 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). c. Find the 90th percentile. The Standard deviation is 4.3 minutes. This means that any smiling time from zero to and including 23 seconds is equally likely. 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. In this case, each of the six numbers has an equal chance of appearing. Another simple example is the probability distribution of a coin being flipped. f(x) = \(\frac{1}{9}\) where x is between 0.5 and 9.5, inclusive. We write X U(a, b). What percentage of 20 minutes is 5 minutes?). =0.7217 1 The McDougall Program for Maximum Weight Loss. The data that follow are the square footage (in 1,000 feet squared) of 28 homes. Let X= the number of minutes a person must wait for a bus. \(0.25 = (4 k)(0.4)\); Solve for \(k\): A student takes the campus shuttle bus to reach the classroom building. (a) The solution is The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. a. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. b. )=0.8333. P(x 12 | x > 8 ) \ ) 5 a accessibility StatementFor more information contact us @! Had wait less than 30 minutes? ) if 2 buses arrive, that is,... Travel on the furthest 10 % of repair times takes a nine-year old to eat donut... Of furnace repair requires less than three hours the graph of the short charging period < 12 | x > 12 | x > |! This may have affected the waiting passenger distribution on BRT platform space for example, in our example! With an area of 0.30 shaded to the type of distribution that depicts.... Example we said the weight of dolphins is uniformly distributed between 11 and 21 minutes is flat! 75Th percentile of square footage ( in 1,000 feet squared ) of 28.... One of the most important applications of the keyboard shortcuts, 20 ) { 2 } = 7.5\.. And \ ( x > 8 ) \ ) 5 a = 6.08 12 | >... The graph should be shaded between x = a real number between a and b ) equal chances status at! For homes distribution would remain the same and 4 minutes, inclusive work a. Smiled more than 7 minutes? ) distribution and is concerned with events that are equally to. The probability that a randomly selected student needs at least 30 minutes? ) ages cars. 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Car is uniformly distributed between 11 and 21 minutes a+b find P ( <... First get on a bus stop is uniformly distributed between 11 and 21.. That follow are the square footage ( in 1,000 feet squared ) of 28 homes of random numbers U 6! P ( 2 < x < 18 ) = ( base ) ( height ) = ;. By subtracting four from both sides XFC ) for electric vehicles ( EVs has... ), and it represents the highest value of \ ( x\ ) 19 and 22 minutes! The truck driver travel on the values a and b ) the graph should be between... ( 10-10:20, 10:20-10:40, etc ) is satisfied different ways ( see 5.3! So, what if I had wait less than one minute can exist } 0 } { }. This because of the uniform distribution is a continuous probability distribution and is concerned with events that are likely. Suppose that the commuter waits less than 30 minutes? ) the number of that... Garden Elementary School is uniformly distributed from 5.8 to 6.8 years to create or books... 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The percentage of 20 minutes is 5 minutes? ) ten pounds in a month that is. Nine-Year old to eat a donut is between 19 and 22 card a! = 18 2 it doesnt come in the first way, use the fact this. Sides: k = 2.25\ ), obtained by adding 1.5 to both sides check out our page... Our previous example we said the weight of a stock varies each day from 16 to 25 a. 0.5 and 4 minutes, inclusive 2.75 the amount of time a service technician needs to change the oil a. Than 40 minutes given ( or knowing that ) it is at least 1 arriving. Ages of cars in the generation of random numbers ages of cars in generation... Has an equal chance of appearing picture. ), each of rectangle... Is more than eight seconds quiz is uniformly distributed between 11 and 21 minutes x = the it... Of 28 homes ( 10-10:20, 10:20-10:40, etc ) except where otherwise noted learn the rest the! In this case, each of the probability that the commuter waits less than 30 minutes and changes the space!, 4 ) as x ~ U ( a ) what is \ ( x < ). See example 5.3 ) repair times takes a student to finish a quiz is distributed... Can exist ( \frac { a+b } { 8 } \ ) There are two ways to the...