14.6 - Uniform Distributions. The number of miles driven by a truck driver falls between 300 and 700, and follows a uniform distribution. The data in Table 5.1 are 55 smiling times, in seconds, of an eight-week-old baby. What is P(2 < x < 18)? Answer: (Round to two decimal place.) They can be said to follow a uniform distribution from one to 53 (spread of 52 weeks). In statistics, uniform distribution is a term used to describe a form of probability distribution where every possible outcome has an equal likelihood of happening. 15 c. This probability question is a conditional. The lower value of interest is 17 grams and the upper value of interest is 19 grams. The probability a person waits less than 12.5 minutes is 0.8333. b. = How likely is it that a bus will arrive in the next 5 minutes? Let \(X =\) the time needed to change the oil in a car. Find the indicated p. View Answer The waiting times between a subway departure schedule and the arrival of a passenger are uniformly. Uniform Distribution between 1.5 and 4 with an area of 0.30 shaded to the left, representing the shortest 30% of repair times. = \(X \sim U(0, 15)\). Find the upper quartile 25% of all days the stock is above what value? The distribution is ______________ (name of distribution). = )( 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 \(\frac{4}{5}\). Find the probability that a randomly selected furnace repair requires more than two hours. For the second way, use the conditional formula from Probability Topics with the original distribution \(X \sim U(0, 23)\): \(P(\text{A|B}) = \frac{P(\text{A AND B})}{P(\text{B})}\). Find P(X<12:5). X is continuous. for 8 < x < 23, P(x > 12|x > 8) = (23 12) Another simple example is the probability distribution of a coin being flipped. The Standard deviation is 4.3 minutes. Uniform Distribution. (a) What is the probability that the individual waits more than 7 minutes? 0+23 Find probability that the time between fireworks is greater than four seconds. As an Amazon Associate we earn from qualifying purchases. The 90th percentile is 13.5 minutes. Note that the shaded area starts at x = 1.5 rather than at x = 0; since X ~ U (1.5, 4), x can not be less than 1.5. Then x ~ U (1.5, 4). Sketch the graph, and shade the area of interest. 1. The data that follow are the square footage (in 1,000 feet squared) of 28 homes. 15+0 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. For the first way, use the fact that this is a conditional and changes the sample space. Random sampling because that method depends on population members having equal chances. The standard deviation of \(X\) is \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}}\). The probability a person waits less than 12.5 minutes is 0.8333. b. 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. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. Then \(X \sim U(0.5, 4)\). The data follow a uniform distribution where all values between and including zero and 14 are equally likely. 1 11 12 (ba) What is the 90th . We write X U(a, b). The Uniform Distribution by OpenStaxCollege is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted. Correct me if I am wrong here, but shouldn't it just be P(A) + P(B)? 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. (In other words: find the minimum time for the longest 25% of repair times.) 1 Find the value \(k\) such that \(P(x < k) = 0.75\). If we get to the bus stop at a random time, the chances of catching a very large waiting gap will be relatively small. \(P(x > k) = (\text{base})(\text{height}) = (4 k)(0.4)\) 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. 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. 0.90 \(P(x < 3) = (\text{base})(\text{height}) = (3 1.5)(0.4) = 0.6\). Lowest value for \(\overline{x}\): _______, Highest value for \(\overline{x}\): _______. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. Want to create or adapt books like this? 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. Department of Earth Sciences, Freie Universitaet Berlin. (230) 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. (Recall: The 90th percentile divides the distribution into 2 parts so. 23 Answer: (Round to two decimal places.) P(x>8) Write the answer in a probability statement. Lets suppose that the weight loss is uniformly distributed. (a) The solution is You already know the baby smiled more than eight seconds. 5 23 X = a real number between a and b (in some instances, X can take on the values a and b). Continuous Uniform Distribution Example 2 The Uniform Distribution. )=0.8333. Write the probability density function. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. State the values of a and \(b\). P(x>12) X ~ U(0, 15). Suppose that the value of a stock varies each day from 16 to 25 with a uniform distribution. 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. What percentile does this represent? 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. Sketch a graph of the pdf of Y. b. 2 ) a. Find the probability that a person is born at the exact moment week 19 starts. Sketch and label a graph of the distribution. Sketch the graph of the probability distribution. a+b f(x) = \(\frac{1}{b-a}\) for a x b. 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. You must reduce the sample space. )=0.90 Use the following information to answer the next three exercises. In real life, analysts use the uniform distribution to model the following outcomes because they are uniformly distributed: Rolling dice and coin tosses. Let X = length, in seconds, of an eight-week-old baby's smile. P(x>2ANDx>1.5) For this example, \(X \sim U(0, 23)\) and \(f(x) = \frac{1}{23-0}\) for \(0 \leq X \leq 23\). Plume, 1995. Solution Let X denote the waiting time at a bust stop. Find the mean, \(\mu\), and the standard deviation, \(\sigma\). 1 2 11 )=20.7 Uniform Distribution between 1.5 and 4 with an area of 0.25 shaded to the right representing the longest 25% of repair times. The data follow a uniform distribution where all values between and including zero and 14 are equally likely. The sample mean = 11.49 and the sample standard deviation = 6.23. 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. P(x>12ANDx>8) What is the probability density function? As waiting passengers occupy more platform space than circulating passengers, evaluation of their distribution across the platform is important. P(x>8) 15 This module describes the properties of the Uniform Distribution which describes a set of data for which all aluesv have an equal probabilit.y Example 1 . The lower value of interest is 0 minutes and the upper value of interest is 8 minutes. \(\mu = \frac{a+b}{2} = \frac{15+0}{2} = 7.5\). Possible waiting times are along the horizontal axis, and the vertical axis represents the probability. However, if you favored short people or women, they would have a higher chance of being given the $100 bill than the other passersby. e. \(\mu =\frac{a+b}{2}\) and \(\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}\), \(\mu =\frac{1.5+4}{2}=2.75\) What is the probability that the waiting time for this bus is less than 5.5 minutes on a given day? Develop analytical superpowers by learning how to use programming and data analytics tools such as VBA, Python, Tableau, Power BI, Power Query, and more. Let x = the time needed to fix a furnace. b. 2.5 Find the probability that a randomly selected furnace repair requires more than two hours. What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? = 7.5. 23 15+0 Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. What percentage of 20 minutes is 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. What is the average waiting time (in minutes)? Therefore, each time the 6-sided die is thrown, each side has a chance of 1/6. P(x 12|x > 8) There are two ways to do the problem. P(x > k) = 0.25 1 For the first way, use the fact that this is a conditional and changes the sample space. 1 Theres only 5 minutes left before 10:20. ) The percentage of the probability is 1 divided by the total number of outcomes (number of passersby). 23 f ( x) = 1 12 1, 1 x 12 = 1 11, 1 x 12 = 0.0909, 1 x 12. = ) The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data are inclusive or exclusive of endpoints. 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. Find the probability that he lost less than 12 pounds in the month. \(f(x) = \frac{1}{4-1.5} = \frac{2}{5}\) for \(1.5 \leq x \leq 4\). )( Write the probability density function. Use the following information to answer the next eight exercises. 2 Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. In this framework (see Fig. The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution. 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. a. On the average, a person must wait 7.5 minutes. \(0.625 = 4 k\), Draw a graph. If you arrive at the bus stop, what is the probability that the bus will show up in 8 minutes or less? When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. What is the probability that the rider waits 8 minutes or less? obtained by dividing both sides by 0.4 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 cumulative distribution function of X is P(X x) = \(\frac{x-a}{b-a}\). 12 = 4.3. A subway train on the Red Line arrives every eight minutes during rush hour. You must reduce the sample space. For this problem, \(\text{A}\) is (\(x > 12\)) and \(\text{B}\) is (\(x > 8\)). What is P(2 < x < 18)? For this problem, A is (x > 12) and B is (x > 8). 0.90=( The mean of \(X\) is \(\mu = \frac{a+b}{2}\). 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. The Sky Train from the terminal to the rentalcar and longterm parking center is supposed to arrive every eight 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. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. To me I thought I would just take the integral of 1/60 dx from 15 to 30, but that is not correct. What is the theoretical standard deviation? How do these compare with the expected waiting time and variance for a single bus when the time is uniformly distributed on \({\rm{(0,5)}}\)? The probability that a randomly selected nine-year old child eats a donut in at least two minutes is _______. 230 Legal. 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