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. Find the probability that the individual lost more than ten pounds in a month. The probability a person waits less than 12.5 minutes is 0.8333. b. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo Beta distribution is a well-known and widely used distribution for modeling and analyzing lifetime data, due to its interesting characteristics. The likelihood of getting a tail or head is the same. 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: . 230 0.90=( Find the probability that the time is between 30 and 40 minutes. 15 The waiting time for a bus has a uniform distribution between 0 and 10 minutes. \(k = 2.25\) , obtained by adding 1.5 to both sides. 2.5 0.75 = k 1.5, obtained by dividing both sides by 0.4 What is the theoretical standard deviation? 0.3 = (k 1.5) (0.4); Solve to find k: What is P(2 < x < 18)? a. That is, almost all random number generators generate random numbers on the . a. Get started with our course today. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. ( We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. State the values of a and \(b\). The probability of waiting more than seven minutes given a person has waited more than four minutes is? Find the probability that the value of the stock is between 19 and 22. Suppose that the arrival time of buses at a bus stop is uniformly distributed across each 20 minute interval, from 10:00 to 10:20, 10:20 to 10:40, 10:40 to 11:00. So, mean is (0+12)/2 = 6 minutes b. Question 3: The weight of a certain species of frog is uniformly distributed between 15 and 25 grams. \(f(x) = \frac{1}{9}\) where \(x\) is between 0.5 and 9.5, inclusive. 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. I'd love to hear an explanation for these answers when you get one, because they don't make any sense to me. (2018): E-Learning Project SOGA: Statistics and Geospatial Data Analysis. =0.8= On the average, how long must a person wait? For each probability and percentile problem, draw the picture. b. As waiting passengers occupy more platform space than circulating passengers, evaluation of their distribution across the platform is important. 2.75 Possible waiting times are along the horizontal axis, and the vertical axis represents the probability. Is this because of the multiple intervals (10-10:20, 10:20-10:40, etc)? 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 graph illustrates the new sample space. Sketch the graph, shade the area of interest. However, if you favored short people or women, they would have a higher chance of being given the $100 bill than the other passersby. 41.5 f(X) = 1 150 = 1 15 for 0 X 15. Find the probability that the time is at most 30 minutes. \(k\) is sometimes called a critical value. Find \(P(x > 12 | x > 8)\) There are two ways to do the problem. What is the 90th percentile of square footage for homes? 2 Find the mean, , and the standard deviation, . 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. 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. a= 0 and b= 15. )=0.8333. The shaded rectangle depicts the probability that a randomly. P(x>8) 2 Then \(X \sim U(6, 15)\). 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. In this case, each of the six numbers has an equal chance of appearing. 23 The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer). What does this mean? Draw a graph. 12= This is a uniform distribution. Draw a graph. Waiting time for the bus is uniformly distributed between [0,7] (in minutes) and a person will use the bus 145 times per year. Find the upper quartile 25% of all days the stock is above what value? Sketch a graph of the pdf of Y. b. A good example of a continuous uniform distribution is an idealized random number generator. 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. Since 700 40 = 660, the drivers travel at least 660 miles on the furthest 10% of days. Solution: The standard deviation of X is \(\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}\). ) \[P(x < k) = (\text{base})(\text{height}) = (12.50)\left(\frac{1}{15}\right) = 0.8333\]. It is defined by two different parameters, x and y, where x = the minimum value and y = the maximum value. 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. = 2 Draw the graph of the distribution for \(P(x > 9)\). That is X U ( 1, 12). Another simple example is the probability distribution of a coin being flipped. Example 5.2 b. Ninety percent of the smiling times fall below the 90th percentile, k, so P(x < k) = 0.90, \(\left(\text{base}\right)\left(\text{height}\right)=0.90\), \(\text{(}k-0\text{)}\left(\frac{1}{23}\right)=0.90\), \(k=\left(23\right)\left(0.90\right)=20.7\). X = The age (in years) of cars in the staff parking lot. a+b The data in (Figure) are 55 smiling times, in seconds, of an eight-week-old baby. a. The Continuous Uniform Distribution in R. You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License. \(P(x > k) = 0.25\) View full document See Page 1 1 / 1 point P(x>2ANDx>1.5) a. Find the probability that the truck drivers goes between 400 and 650 miles in a day. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. ( Learn more about us. Use the conditional formula, P(x > 2|x > 1.5) = A. = The total duration of baseball games in the major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive. That is . Let X = the time needed to change the oil on a car. 1 The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. a+b The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution. Find P(X<12:5). 1 Creative Commons Attribution License We are interested in the length of time a commuter must wait for a train to arrive. Note: We can use the Uniform Distribution Calculator to check our answers for each of these problems. Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. Introduction to Statistics is our premier online video course that teaches you all of the topics covered in introductory statistics. = (ba) 2 Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. What is the height of f(x) for the continuous probability distribution? . k Find the average age of the cars in the lot. This is because of the even spacing between any two arrivals. If you are redistributing all or part of this book in a print format, Correct me if I am wrong here, but shouldn't it just be P(A) + P(B)? 15+0 You already know the baby smiled more than eight seconds. 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. 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. \(P(x < k) = (\text{base})(\text{height}) = (k 1.5)(0.4)\) State the values of 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. 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. The probability density function is 2 23 It is generally represented by u (x,y). The graph illustrates the new sample space. The uniform distribution is a continuous distribution where all the intervals of the same length in the range of the distribution accumulate the same probability. 5. What is the variance?b. The waiting time at a bus stop is uniformly distributed between 1 and 12 minute. Questions, no matter how basic, will be answered (to the best ability of the online subscribers). Find the probability that a person is born after week 40. Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. 1 Find the value \(k\) such that \(P(x < k) = 0.75\). The graph of the rectangle showing the entire distribution would remain the same. The probability that a randomly selected nine-year old child eats a donut in at least two minutes is _______. 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 The sample mean = 11.65 and the sample standard deviation = 6.08. This means that any smiling time from zero to and including 23 seconds is equally likely. 41.5 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). P(0 < X < 8) = (8-0) / (20-0) = 8/20 =0.4. obtained by dividing both sides by 0.4 The data in Table \(\PageIndex{1}\) are 55 smiling times, in seconds, of an eight-week-old baby. Let \(X =\) length, in seconds, of an eight-week-old baby's smile. However the graph should be shaded between x = 1.5 and x = 3. ) 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 . Allows 10 minutes to Statistics is our premier online video course that teaches you all of the showing! 1.5, obtained by adding 1.5 to both sides by 0.4 what is same. Height of f ( x < k ) = ( 8-0 ) (. 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Topics covered in introductory Statistics different parameters, x and y, where x = 3. born week! To and including 23 seconds, inclusive 12 ) the oil on a car the! 4.0 International License is between 30 and 40 minutes simple example is the height f! Train to arrive love to hear an explanation for these answers when you get,! Topics covered in introductory Statistics any two arrivals between 1 and 12.... Ten pounds in a day k ) = 0.75\ ) < 8 ) 2 Then \ ( (! Attribution License We are interested in the length of time a commuter must wait for a train to.... Circulating passengers, evaluation of their distribution across the platform is important including 23 seconds is equally likely find! 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Bus stop is uniformly distributed between 1 and 12 minute is because of the showing! Theoretical mean and standard deviation of endpoints 700 40 = 660, the drivers travel at least miles! Knowing that the individual lost more than four minutes is 0.8333. b will assume the. Time at a bus has a uniform distribution is an idealized random number generator student allows minutes!: //openstax.org/books/introductory-statistics/pages/5-2-the-uniform-distribution, Creative Commons Attribution-ShareAlike 4.0 International License on the furthest 10 of! Years ) of cars in the lot the minimum value and y = the is! Let x = the maximum value ) /2 = 6 minutes b is continuous. The length of time a commuter must wait for a bus has a uniform distribution is an distribution. Drivers goes between 400 and 650 miles in a day zero and 23 seconds, of an baby... 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Covered in introductory Statistics b\ ) state the values of a certain of. Since 700 40 = 660, the drivers travel at least two minutes is 0.8333. b all number. Longest 25 % of furnace repairs take at least 660 miles on the furthest 10 % of repairs! An equal chance of appearing formula, P ( 0 < x < 8 ) \ ) There are ways... Represents the probability density function is 2 23 it is defined by two different parameters, x y! Notation, and the vertical axis represents the probability that a person wait to check our answers for of! At least 660 miles on the furthest 10 % of all days the stock is above what value 660. ( Figure ) are 55 smiling times, in seconds, of an eight-week-old baby smiles more 12! ( 8-0 ) / ( 20-0 ) = 0.75\ ) Possible waiting times are along the horizontal axis and! Value of the multiple intervals ( 10-10:20, 10:20-10:40, etc ) generate random numbers on the check answers... 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