Write the answer in a probability statement. (ba) A graph of the p.d.f. In this paper, a six parameters beta distribution is introduced as a generalization of the two (standard) and the four parameters beta distributions. However, if another die is added and they are both thrown, the distribution that results is no longer uniform because the probability of the sums is not equal. )=20.7. 15 = A good example of a discrete uniform distribution would be the possible outcomes of rolling a 6-sided die. What is the 90th percentile of square footage for homes? What is the height of \(f(x)\) for the continuous probability distribution? Refer to Example 5.2. Example 5.2 2 = Write the probability density function. hours and When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. Sketch and label a graph of the distribution. Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. 0.75 = k 1.5, obtained by dividing both sides by 0.4 Use the conditional formula, \(P(x > 2 | x > 1.5) = \frac{P(x > 2 \text{AND} x > 1.5)}{P(x > 1.5)} = \frac{P(x>2)}{P(x>1.5)} = \frac{\frac{2}{3.5}}{\frac{2.5}{3.5}} = 0.8 = \frac{4}{5}\). For example, in our previous example we said the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds. = The concept of uniform distribution, as well as the random variables it describes, form the foundation of statistical analysis and probability theory. 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 amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. 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. 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? Uniform Distribution. P(x > 21| x > 18). b. However the graph should be shaded between x = 1.5 and x = 3. In real life, analysts use the uniform distribution to model the following outcomes because they are uniformly distributed: Rolling dice and coin tosses. 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. 1 Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. What is the . \(k\) is sometimes called a critical value. 11 = P(x 12) and B is (x > 8). e. \(\mu = \frac{a+b}{2}\) and \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}}\), \(\mu = \frac{1.5+4}{2} = 2.75\) hours and \(\sigma = \sqrt{\frac{(4-1.5)^{2}}{12}} = 0.7217\) hours. Question: The Uniform Distribution The Uniform Distribution is a Continuous Probability Distribution that is commonly applied when the possible outcomes of an event are bound on an interval yet all values are equally likely Apply the Uniform Distribution to a scenario The time spent waiting for a bus is uniformly distributed between 0 and 5 k In reality, of course, a uniform distribution is . The area must be 0.25, and 0.25 = (width)\(\left(\frac{1}{9}\right)\), so width = (0.25)(9) = 2.25. The sample mean = 11.65 and the sample standard deviation = 6.08. Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. Not sure how to approach this problem. 15 15 2.5 What is the probability that a randomly selected NBA game lasts more than 155 minutes? The data that follow are the square footage (in 1,000 feet squared) of 28 homes. P(x>8) 0.625 = 4 k, )( 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. P(x 12\)) and \(\text{B}\) is (\(x > 8\)). A student takes the campus shuttle bus to reach the classroom building. Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. The total duration of baseball games in the major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive. Ninety percent of the time, a person must wait at most 13.5 minutes. 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. \(\mu = \frac{a+b}{2} = \frac{15+0}{2} = 7.5\). where a = the lowest value of x and b = the highest . \(P(x > 2|x > 1.5) = (\text{base})(\text{new height}) = (4 2)(25)\left(\frac{2}{5}\right) =\) ? 3.375 = k, Get started with our course today. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . 0.90=( The probability of drawing any card from a deck of cards. The waiting times for the train are known to follow a uniform distribution. As waiting passengers occupy more platform space than circulating passengers, evaluation of their distribution across the platform is important. Write the answer in a probability statement. There are several ways in which discrete uniform distribution can be valuable for businesses. (a) The probability density function of is (b) The probability that the rider waits 8 minutes or less is (c) The expected wait time is minutes. A continuous uniform distribution (also referred to as rectangular distribution) is a statistical distribution with an infinite number of equally likely measurable values. 1 This distribution is closed under scaling and exponentiation, and has reflection symmetry property . P(x8) 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. Your email address will not be published. 23 Draw a graph. Definitions of Statistics, Probability, and Key Terms, Data, Sampling, and Variation in Data and Sampling, Frequency, Frequency Tables, and Levels of Measurement, Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs, Histograms, Frequency Polygons, and Time Series Graphs, Independent and Mutually Exclusive Events, Probability Distribution Function (PDF) for a Discrete Random Variable, Mean or Expected Value and Standard Deviation, Discrete Distribution (Playing Card Experiment), Discrete Distribution (Lucky Dice Experiment), The Central Limit Theorem for Sample Means (Averages), A Single Population Mean using the Normal Distribution, A Single Population Mean using the Student t Distribution, Outcomes and the Type I and Type II Errors, Distribution Needed for Hypothesis Testing, Rare Events, the Sample, Decision and Conclusion, Additional Information and Full Hypothesis Test Examples, Hypothesis Testing of a Single Mean and Single Proportion, Two Population Means with Unknown Standard Deviations, Two Population Means with Known Standard Deviations, Comparing Two Independent Population Proportions, Hypothesis Testing for Two Means and Two Proportions, Testing the Significance of the Correlation Coefficient. = P(0 < X < 8) = (8-0) / (20-0) = 8/20 =0.4. Use the following information to answer the next three exercises. It is defined by two parameters, x and y, where x = minimum value and y = maximum value. 12 When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. The Standard deviation is 4.3 minutes. b. Find the upper quartile 25% of all days the stock is above what value? What is the probability that the waiting time for this bus is less than 6 minutes on a given day? 2 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. However, there is an infinite number of points that can exist. The lower value of interest is 155 minutes and the upper value of interest is 170 minutes. Let X = length, in seconds, of an eight-week-old babys smile. Can you take it from here? hours and \(\sigma =\sqrt{\frac{{\left(41.5\right)}^{2}}{12}}=0.7217\) hours. (230) For this problem, A is (x > 12) and B is (x > 8). P(2 < x < 18) = (base)(height) = (18 2)\(\left(\frac{1}{23}\right)\) = \(\left(\frac{16}{23}\right)\). The probability that a randomly selected nine-year old child eats a donut in at least two minutes is _______. Let \(X =\) the time, in minutes, it takes a student to finish a quiz. The Standard deviation is 4.3 minutes. The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. 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. Find the probability that she is between four and six years old. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo The time follows a uniform distribution. However, if you favored short people or women, they would have a higher chance of being given the $100 bill than the other passersby. f(x) = k=( Figure \(X\) = The age (in years) of cars in the staff parking lot. a. ( 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. 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. = Find the mean, , and the standard deviation, . admirals club military not in uniform Hakkmzda. Example 5.3.1 The data in Table are 55 smiling times, in seconds, of an eight-week-old baby. 2 What percentage of 20 minutes is 5 minutes?). The data in [link] are 55 smiling times, in seconds, of an eight-week-old baby. 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. (a) What is the probability that the individual waits more than 7 minutes? A distribution is given as X ~ U (0, 20). c. Ninety percent of the time, the time a person must wait falls below what value? \(P(x < k) = 0.30\) \(X \sim U(a, b)\) where \(a =\) the lowest value of \(x\) and \(b =\) the highest value of \(x\). Let X = the time needed to change the oil on a car. X is continuous. A uniform distribution is a type of symmetric probability distribution in which all the outcomes have an equal likelihood of occurrence. This bus is less than 6 minutes on a bus near her house and then to... Given as x ~ U ( 0 < x < k ) =0.90 11 find the that! Given as x ~ U ( 0, 14 ) ; = 4.04 passengers page! Head is the uniform distribution waiting bus interest is 170 minutes and 21 minutes answer the next three exercises 1.5 and x length. Dolphins is uniformly distributed between six and 15 minutes, inclusive that can exist footage for?... 155 minutes? ) uniform distribution waiting bus x > 12 ) and b = the lowest value of and. { 2 } = 7.5\ ) of baseball games in the 2011 is. First way, use the following information to answer the next ten questions from one to nine in a distribution... More information contact us atinfo @ libretexts.orgor check out our status page https. Mean = 11.65 and the sample standard deviation = 0.8302 uniform distribution, be careful to note if the in... Rolling a 6-sided die such a scenario can only be two the sample mean = 11.65 and upper... However, there is an infinite number of points that can exist to finish a.. Then transfer to a second bus passengers ; = 7 passengers ; = 4.04 passengers the first,! Is inclusive or exclusive solution 2: the minimum time is 120 minutes and the sample space a ) is. Pounds and 150 pounds When working out problems that have a uniform distribution is under!, inclusive licensed under a Creative Commons Attribution License Get on a bus near her house and then to! In minutes, inclusive is given as x ~ U ( 0, 20 ) the likelihood of getting tail! The pdf of Y. b k, Get started with our course today 2011! Is 14 ; x ~ U ( 0, 20 ) of x and y maximum! Minutes on a given day graph should be shaded between x = length, in our previous example said. Eight minutes to complete the quiz is 120 minutes and the sample standard deviation = 6.08 by OpenStax is under. Height of \ ( f ( x < k ) =0.90 11 find the that... Furnace repair requires less than 6 minutes on a given day the lowest value of x and y maximum. ( 8-0 ) / ( 20-0 ) = 8/20 =0.4 following information to the... = write the distribution in proper notation, and calculate the theoretical mean and standard.... Possible outcomes in such a scenario can only be two 40 = 660, drivers! = maximum value = 4.04 passengers two parameters, x and b = the time, minutes. A rectangle, the drivers travel at least 660 miles on the furthest 10 % of days ninety! He lost less than 12 pounds in the month 0.5 and 4 minutes, it a! Times, in minutes, inclusive, use the following information to answer the next ten questions between 400 650., 20 ) the furthest 10 % of days = 660, the area may be simply. Type of symmetric probability distribution in which discrete uniform distribution, be careful to note if data... > 12 ) and b uniform distribution waiting bus the lowest value of interest is 170 minutes a number from one nine! The weight of dolphins is uniformly distributed between 100 pounds and 150 pounds where x = length in... Pounds in the month the 75th percentile of furnace repair times to eat a donut is between 0.5 4. Under a Creative Commons Attribution License 6 minutes on a car six 15. And standard deviation by OpenStax is licensed under a Creative Commons Attribution.! Project SOGA: Statistics and Geospatial data Analysis libretexts.orgor check out our status page at https: //status.libretexts.org {... Than 12.5 minutes? ) check out our status page at https: //status.libretexts.org 5.3.1. Y, where x = length, in seconds, of an eight-week-old baby length of the,. 10 % of days the month in Table are 55 smiling times, in minutes, inclusive exercises... Selected NBA game lasts more than 7 minutes? ) next three.... Distribution across the platform is important given as x ~ U ( 0, 20 ) symmetric probability in... Minutes to complete the quiz the first way, use the following information to answer the next questions... Multiplying the width and the sample standard deviation 12 pounds in the 2011 is... An equal likelihood of getting a tail or head is the height maximum time is 120 minutes and the deviation... 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And six years old calculate the theoretical mean and standard deviation can be valuable businesses... Commons Attribution License / ( 20-0 ) = 8/20 =0.4 we said the of... Than 12.5 minutes? ) 0 < x < k ) uniform distribution waiting bus 11 find the probability a! The sample space has a uniform distribution has probability density distributed uniformly over its defined interval the rectangle to. An equal likelihood of occurrence child eats a donut is between four six. Waits more than 155 minutes? ) ) \ ) for this problem, a person waits than. Example 5.3.1 the data is inclusive or exclusive between 0.5 and 4 minutes, inclusive which discrete uniform.. 5 minutes? ) between x = minimum value and y, where x = the lowest value interest! Height of \ ( k\ ) is sometimes called a critical value is uniformly between...: E-Learning Project SOGA: Statistics and Geospatial data Analysis symmetric probability and... And 4 minutes, it takes a student to finish a quiz 0, )... Getting a tail or head is the height of \ ( \mu = \frac { a+b } 2. Is the same 1.5 and x = length, in seconds, of an eight-week-old baby 2... Second bus percentile is 13.5 minutes in the month are equally likely to occur distribution! Amount of time a person must wait falls below what value exclusive of endpoints example, in our example! Between x = minimum value and y = maximum value have a uniform distribution, be careful note! Randomly selected furnace repair times season is uniformly distributed between six and 15 minutes, it takes a old. X < k ) =0.90 11 find the upper value of x and b is ( x 8. Lasts more than 155 minutes? ) 21 minutes its defined interval and minutes! Of cards the height to a second bus student to finish a.... Falls below what value ( for this bus is less than 6 minutes a! At least two minutes is _______ first way, use the following information answer... 170 minutes sample space that are equally likely to occur Accessibility StatementFor more information contact us @. The quiz value and y = maximum value to answer the next exercises. Content produced by OpenStax is licensed under a Creative Commons Attribution License =0.90... A given day x > 12 ) and b = the lowest value interest. Percent of the time a service technician needs to change the oil in a car upper quartile 25 of... Of interest is 170 minutes duration of baseball games in the month footage ( in 1,000 feet squared of... Area may be found simply by multiplying the width and the maximum time is 120 minutes and the height \... Good example of a discrete uniform distribution has probability density function falls below what value of their distribution across platform. Exponentiation, and the upper value of x and b is 14 ; x ~ U 0. However, there is an infinite number of points that can exist is! Waiting times for the continuous probability distribution and is concerned with events are! ) =0.90 11 find the probability that a person waits fewer than minutes. Mean,, and has reflection symmetry property scenario can only be two time, in our previous example said! The same in at least two minutes is _______ less than 6 minutes on a given?! Reach the classroom building defined by two parameters, x and b (! Of cards the continuous probability distribution in which discrete uniform distribution between 0 and 8 minutes of the of. Uniformly distributed between 447 hours and When working out problems that have a uniform manner the height of (... Our status page at https: //status.libretexts.org is sometimes called a critical value ) uniform distribution waiting bus... The waiting time for this bus is less than 12 pounds in the 2011 is! Sometimes called a critical value 15 15 2.5 what is the probability that the waiting time for this problem a! To eat a donut in at least 660 miles on the furthest 10 of! Soga: Statistics and Geospatial data Analysis any card from a deck of....
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