f(x) = \(\frac{1}{b-a}\) for a x b. A bus arrives at a bus stop every 7 minutes. Question 2: The length of an NBA game is uniformly distributed between 120 and 170 minutes. (Hint the if it comes in the first 10 minutes and the last 15 minutes, it must come within the 5 minutes of overlap from 10:05-10:10. 1 The Sky Train from the terminal to the rentalcar and longterm parking center is supposed to arrive every eight minutes. We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. 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). What is the probability that a person waits fewer than 12.5 minutes? Draw a graph. f(x) = The graph illustrates the new sample space. Find the mean and the standard deviation. Find the probability that the time is more than 40 minutes given (or knowing that) it is at least 30 minutes. (a) The probability density function of X is. = uniform distribution, in statistics, distribution function in which every possible result is equally likely; that is, the probability of each occurring is the same. What is the probability that the rider waits 8 minutes or less? The mean of uniform distribution is (a+b)/2, where a and b are limits of the uniform distribution. What percentile does this represent? c. Find the 90th percentile. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. 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. looks like this: f (x) 1 b-a X a b. \(a =\) smallest \(X\); \(b =\) largest \(X\), The standard deviation is \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}}\), Probability density function: \(f(x) = \frac{1}{b-a} \text{for} a \leq X \leq b\), Area to the Left of \(x\): \(P(X < x) = (x a)\left(\frac{1}{b-a}\right)\), Area to the Right of \(x\): 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 \(a < x < b\). Find the probability that a randomly selected furnace repair requires less than three hours. Solve the problem two different ways (see [link]). Let X = the number of minutes a person must wait for a bus. = . A good example of a continuous uniform distribution is an idealized random number generator. Find the probability that she is over 6.5 years old. View full document See Page 1 1 / 1 point Let X = the time, in minutes, it takes a nine-year old child to eat a donut. A continuous probability distribution is a Uniform distribution and is related to the events which are equally likely to occur. 11 k=( Darker shaded area represents P(x > 12). \(P(x > 2|x > 1.5) = (\text{base})(\text{new height}) = (4 2)(25)\left(\frac{2}{5}\right) =\) ? a+b Note that the shaded area starts at \(x = 1.5\) rather than at \(x = 0\); since \(X \sim U(1.5, 4)\), \(x\) can not be less than 1.5. P(x>2) 0.25 = (4 k)(0.4); Solve for k: The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. = \(\frac{6}{9}\) = \(\frac{2}{3}\). Refer to Example 5.3.1. Sixty percent of commuters wait more than how long for the train? 2 Not all uniform distributions are discrete; some are continuous. ) 1 \(P(2 < x < 18) = (\text{base})(\text{height}) = (18 2)\left(\frac{1}{23}\right) = \left(\frac{16}{23}\right)\). = 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. 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. = For the second way, use the conditional formula from Probability Topics with the original distribution \(X \sim U(0, 23)\): \(P(\text{A|B}) = \frac{P(\text{A AND B})}{P(\text{B})}\). For the second way, use the conditional formula from Probability Topics with the original distribution X ~ U (0, 23): P(A|B) = \(\frac{P\left(A\text{AND}B\right)}{P\left(B\right)}\). 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 P(x > 12|x > 8) There are two ways to do the problem. = 11.50 seconds and = \(\sqrt{\frac{{\left(23\text{}-\text{}0\right)}^{2}}{12}}\) Write a newf(x): f(x) = \(\frac{1}{23\text{}-\text{8}}\) = \(\frac{1}{15}\), P(x > 12|x > 8) = (23 12)\(\left(\frac{1}{15}\right)\) = \(\left(\frac{11}{15}\right)\). 5 5 The lower value of interest is 155 minutes and the upper value of interest is 170 minutes. The shuttle bus arrives at his stop every 15 minutes but the actual arrival time at the stop is random. What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? Let \(X =\) the time, in minutes, it takes a nine-year old child to eat a donut. (a) What is the probability that the individual waits more than 7 minutes? 1 Uniform Distribution between 1.5 and 4 with an area of 0.30 shaded to the left, representing the shortest 30% of repair times. Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. ) P(x>12) Uniform Distribution between 1.5 and 4 with an area of 0.30 shaded to the left, representing the shortest 30% of repair times. 15 \nonumber\]. = The age of cars in the staff parking lot of a suburban college is uniformly distributed from six months (0.5 years) to 9.5 years. 5.2 The Uniform Distribution. This is a modeling technique that uses programmed technology to identify the probabilities of different outcomes. = 1 The uniform distribution defines equal probability over a given range for a continuous distribution. However the graph should be shaded between \(x = 1.5\) and \(x = 3\). 1.0/ 1.0 Points. 15 What is the theoretical standard deviation? 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. On the average, a person must wait 7.5 minutes. 3.375 hours is the 75th percentile of furnace repair times. f(x) = \(\frac{1}{9}\) where x is between 0.5 and 9.5, inclusive. The age of cars in the staff parking lot of a suburban college is uniformly distributed from six months (0.5 years) to 9.5 years. You will wait for at least fifteen minutes before the bus arrives, and then, 2). 12, For this problem, the theoretical mean and standard deviation are. = You must reduce the sample space. Shade the area of interest. 15 Plume, 1995. ) 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. Let k = the 90th percentile. Find the probability that a person is born at the exact moment week 19 starts. 2 \(0.90 = (k)\left(\frac{1}{15}\right)\) = For the first way, use the fact that this is a conditional and changes the sample space. 15. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. Discrete uniform distribution is also useful in Monte Carlo simulation. This distribution is closed under scaling and exponentiation, and has reflection symmetry property . Thank you! A deck of cards also has a uniform distribution. for 8 < x < 23, P(x > 12|x > 8) = (23 12) A continuous uniform distribution is a statistical distribution with an infinite number of equally likely measurable values. P(x > 2|x > 1.5) = (base)(new height) = (4 2) 2 . A uniform distribution has the following properties: The area under the graph of a continuous probability distribution is equal to 1. The answer for 1) is 5/8 and 2) is 1/3. Solution Let X denote the waiting time at a bust stop. 15 Second way: Draw the original graph for X ~ U (0.5, 4). For each probability and percentile problem, draw the picture. = That is, 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 is \(\frac{4}{5}\). (2018): E-Learning Project SOGA: Statistics and Geospatial Data Analysis. So, mean is (0+12)/2 = 6 minutes b. Define the random . Sketch a graph of the pdf of Y. b. 12 Formulas for the theoretical mean and standard deviation are, \[\sigma = \sqrt{\frac{(b-a)^{2}}{12}} \nonumber\], For this problem, the theoretical mean and standard deviation are, \[\mu = \frac{0+23}{2} = 11.50 \, seconds \nonumber\], \[\sigma = \frac{(23-0)^{2}}{12} = 6.64\, seconds. 1 15 Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . In this case, each of the six numbers has an equal chance of appearing. Therefore, the finite value is 2. Let X = length, in seconds, of an eight-week-old babys smile. Find the probability that a randomly selected furnace repair requires less than three hours. Buses run every 30 minutes without fail, hence the next bus will come any time during the next 30 minutes with evenly distributed probability (a uniform distribution). 0+23 )( The McDougall Program for Maximum Weight Loss. = P(B) \(P(x < k) = (\text{base})(\text{height}) = (k 1.5)(0.4)\) 1). All values \(x\) are equally likely. b. ) For the first way, use the fact that this is a conditional and changes the sample space. 2 In this paper, a six parameters beta distribution is introduced as a generalization of the two (standard) and the four parameters beta distributions. \(f(x) = \frac{1}{9}\) where \(x\) is between 0.5 and 9.5, inclusive. It is defined by two different parameters, x and y, where x = the minimum value and y = the maximum value. Sketch the graph, shade the area of interest. The Standard deviation is 4.3 minutes. To find f(x): f (x) = a. 2 The concept of uniform distribution, as well as the random variables it describes, form the foundation of statistical analysis and probability theory. \(f\left(x\right)=\frac{1}{8}\) where \(1\le x\le 9\). \(0.25 = (4 k)(0.4)\); Solve for \(k\): Extreme fast charging (XFC) for electric vehicles (EVs) has emerged recently because of the short charging period. \(0.625 = 4 k\), 2.75 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 . 1 Let \(X =\) the time needed to change the oil on a car. 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. A good example of a discrete uniform distribution would be the possible outcomes of rolling a 6-sided die. Then \(X \sim U(6, 15)\). 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). This means that any smiling time from zero to and including 23 seconds is equally likely. Then \(x \sim U(1.5, 4)\). \(P(x < 4) =\) _______. Correct answers: 3 question: The waiting time for a bus has a uniform distribution between 0 and 8 minutes. P(2 < x < 18) = (base)(height) = (18 2) Let X = the time needed to change the oil on a car. 3.375 hours is the 75th percentile of furnace repair times. What are the constraints for the values of x? The likelihood of getting a tail or head is the same. 1 Then X ~ U (0.5, 4). ( As one of the simplest possible distributions, the uniform distribution is sometimes used as the null hypothesis, or initial hypothesis, in hypothesis testing, which is used to ascertain the accuracy of mathematical models. Statistics and Probability questions and answers A bus arrives every 10 minutes at a bus stop. Find the 90th percentile for an eight-week-old babys smiling time. What is the probability density function? Write the random variable \(X\) in words. =45 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. a is zero; b is 14; X ~ U (0, 14); = 7 passengers; = 4.04 passengers. 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. Given that the stock is greater than 18, find the probability that the stock is more than 21. The possible outcomes in such a scenario can only be two. = In real life, analysts use the uniform distribution to model the following outcomes because they are uniformly distributed: Rolling dice and coin tosses. A student takes the campus shuttle bus to reach the classroom building. for 1.5 x 4. The notation for the uniform distribution is. Your starting point is 1.5 minutes. Find the 90th percentile. Find the upper quartile 25% of all days the stock is above what value? What is P(2 < x < 18)? Question 12 options: Miles per gallon of a vehicle is a random variable with a uniform distribution from 23 to 47. The interval of values for \(x\) is ______. Random sampling because that method depends on population members having equal chances. What is the average waiting time (in minutes)? Answer: (Round to two decimal places.) 2.1.Multimodal generalized bathtub. The probability that a randomly selected nine-year old child eats a donut in at least two minutes is _______. 5 Find the probability that the time is between 30 and 40 minutes. That is, almost all random number generators generate random numbers on the . Uniform Distribution between 1.5 and four with shaded area between two and four representing the probability that the repair time, Uniform Distribution between 1.5 and four with shaded area between 1.5 and three representing the probability that the repair time. What is the height of \(f(x)\) for the continuous probability distribution? f (x) = \(\frac{1}{15\text{}-\text{}0}\) = \(\frac{1}{15}\) Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. That is . 2 Then X ~ U (0.5, 4). Uniform Distribution between 1.5 and 4 with an area of 0.25 shaded to the right representing the longest 25% of repair times. = 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. 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). for 0 x 15. For this reason, it is important as a reference distribution. k is sometimes called a critical value. In this distribution, outcomes are equally likely. 2.5 (Recall: The 90th percentile divides the distribution into 2 parts so. Want to create or adapt books like this? However, if you favored short people or women, they would have a higher chance of being given the $100 bill than the other passersby. b. Ninety percent of the smiling times fall below the 90th percentile, k, so P(x < k) = 0.90. 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. The Bus wait times are uniformly distributed between 5 minutes and 23 minutes. The percentage of the probability is 1 divided by the total number of outcomes (number of passersby). k=(0.90)(15)=13.5 (b-a)2 Ninety percent of the time, a person must wait at most 13.5 minutes. 2 2 What is the expected waiting time? It is assumed that the waiting time for a particular individual is a random variable with a continuous uniform distribution.
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