2.1.Multimodal generalized bathtub. What is the probability that the waiting time for this bus is less than 5.5 minutes on a given day? Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. 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. 2.5 P(x>12) 2 The data that follow are the number of passengers on 35 different charter fishing boats. Then \(X \sim U(6, 15)\). 3.375 hours is the 75th percentile of furnace repair times. 15 Find the probability that a randomly selected furnace repair requires more than two hours. The waiting time at a bus stop is uniformly distributed between 1 and 12 minute. 15 Let X = length, in seconds, of an eight-week-old baby's smile. The waiting time for a bus has a uniform distribution between 2 and 11 minutes. The sample mean = 11.49 and the sample standard deviation = 6.23. Entire shaded area shows P(x > 8). 5 The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform 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 4545. The probability a person waits less than 12.5 minutes is 0.8333. b. The Uniform Distribution by OpenStaxCollege is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted. The percentage of the probability is 1 divided by the total number of outcomes (number of passersby). What is \(P(2 < x < 18)\)? Discrete uniform distribution is also useful in Monte Carlo simulation. 0.625 = 4 k, e. For this problem, A is (x > 12) and B is (x > 8). obtained by subtracting four from both sides: \(k = 3.375\) 3.5 ( Random sampling because that method depends on population members having equal chances. ) On the average, how long must a person wait? . Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. What is the theoretical standard deviation? When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. 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. That is . This may have affected the waiting passenger distribution on BRT platform space. 150 = \(\frac{0\text{}+\text{}23}{2}\) The time follows a uniform distribution. P(155 < X < 170) = (170-155) / (170-120) = 15/50 = 0.3. 1 Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. The longest 25% of furnace repair times take at least how long? \(X\) = The age (in years) of cars in the staff parking lot. 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. uniform distribution, in statistics, distribution function in which every possible result is equally likely; that is, the probability of each occurring is the same. 11 In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. 12 Find the probability. P(x > 21| x > 18). = Births are approximately uniformly distributed between the 52 weeks of the year. The probability density function of X is \(f\left(x\right)=\frac{1}{b-a}\) for a x b. The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer). The graph of the rectangle showing the entire distribution would remain the same. Correct answers: 3 question: The waiting time for a bus has a uniform distribution between 0 and 8 minutes. Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. Question 1: A bus shows up at a bus stop every 20 minutes. 2 1 0.90=( Unlike discrete random variables, a continuous random variable can take any real value within a specified range. The probability that a randomly selected nine-year old child eats a donut in at least two minutes is _______. The uniform distribution defines equal probability over a given range for a continuous distribution. P(x 2|x > 1.5) = \(\frac{P\left(x>2\text{AND}x>1.5\right)}{P\left(x>\text{1}\text{.5}\right)}=\frac{P\left(x>2\right)}{P\left(x>1.5\right)}=\frac{\frac{2}{3.5}}{\frac{2.5}{3.5}}=\text{0}\text{.8}=\frac{4}{5}\). Solve the problem two different ways (see [link]). = and )( P(A or B) = P(A) + P(B) - P(A and B). What is P(2 < x < 18)? 15 Uniform distribution can be grouped into two categories based on the types of possible outcomes. P(AANDB) Sketch the graph, shade the area of interest. (In other words: find the minimum time for the longest 25% of repair times.) What is the 90th . Find the probability that the truck drivers goes between 400 and 650 miles in a day. In this case, each of the six numbers has an equal chance of appearing. What is the probability that a person waits fewer than 12.5 minutes? Sketch the graph, and shade the area of interest. Find the 90th percentile. For example, we want to predict the following: The amount of timeuntilthe customer finishes browsing and actually purchases something in your store (success). For this example, x ~ U(0, 23) and f(x) = On the average, a person must wait 7.5 minutes. For the first way, use the fact that this is a conditional and changes the sample space. = Question 3: The weight of a certain species of frog is uniformly distributed between 15 and 25 grams. Formulas for the theoretical mean and standard deviation are, = The data follow a uniform distribution where all values between and including zero and 14 are equally likely. = \(\frac{6}{9}\) = \(\frac{2}{3}\). Introduction to Statistics is our premier online video course that teaches you all of the topics covered in introductory statistics. 2 Refer to Example 5.3.1. Solution Let X denote the waiting time at a bust stop. Get started with our course today. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo Therefore, the finite value is 2. The probability density function of \(X\) is \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\). Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. Let \(X =\) the time needed to change the oil on a car. Thank you! 1 A bus arrives at a bus stop every 7 minutes. 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. Note: We can use the Uniform Distribution Calculator to check our answers for each of these problems. You must reduce the sample space. 15. What is the 90th percentile of square footage for homes? (a) What is the probability that the individual waits more than 7 minutes? P(x > k) = (base)(height) = (4 k)(0.4) OR. X ~ U(0, 15). The waiting time for a bus has a uniform distribution between 0 and 8 minutes. Sixty percent of commuters wait more than how long for the train? 1 Find P(x > 12|x > 8) There are two ways to do the problem. You already know the baby smiled more than eight seconds. k=(0.90)(15)=13.5 Learn more about how Pressbooks supports open publishing practices. This is a uniform distribution. Find the mean and the standard deviation. Therefore, each time the 6-sided die is thrown, each side has a chance of 1/6. (b) What is the probability that the individual waits between 2 and 7 minutes? 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. The probability density function is 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. = Let X = the time, in minutes, it takes a student to finish a quiz. Press J to jump to the feed. State the values of a and b. P(x > 2|x > 1.5) = (base)(new height) = (4 2) In words, define the random variable \(X\). Then \(X \sim U(0.5, 4)\). 23 Find the mean and the standard deviation. 15 Example 5.2 23 = Ninety percent of the time, a person must wait at most 13.5 minutes. 15 Then X ~ U (0.5, 4). 12 To find f(x): f (x) = What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? Solution 2: The minimum time is 120 minutes and the maximum time is 170 minutes. 1 How likely is it that a bus will arrive in the next 5 minutes? What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? State the values of a and \(b\). A continuous uniform distribution (also referred to as rectangular distribution) is a statistical distribution with an infinite number of equally likely measurable values. 12 Given that the stock is greater than 18, find the probability that the stock is more than 21. Creative Commons Attribution License In their calculations of the optimal strategy . Find P(x > 12|x > 8) There are two ways to do the problem. The notation for the uniform distribution is. 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. We write \(X \sim U(a, b)\). a. Solution: For example, in our previous example we said the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds. k The Standard deviation is 4.3 minutes. The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. 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 . Use the following information to answer the next ten questions. Let X = the time, in minutes, it takes a student to finish a quiz. What is the probability that a person waits fewer than 12.5 minutes? Question 2: The length of an NBA game is uniformly distributed between 120 and 170 minutes. \(a\) is zero; \(b\) is \(14\); \(X \sim U (0, 14)\); \(\mu = 7\) passengers; \(\sigma = 4.04\) passengers. where a = the lowest value of x and b = the highest . The data that follow are the number of passengers on 35 different charter fishing boats. Suppose it is known that the individual lost more than ten pounds in a month. 1 Answer: (Round to two decimal place.) X is continuous. With continuous uniform distribution, just like discrete uniform distribution, every variable has an equal chance of happening. 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. Ninety percent of the time, a person must wait at most 13.5 minutes. In this framework (see Fig. . That is X U ( 1, 12). 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 . Let \(X =\) length, in seconds, of an eight-week-old baby's smile. Lowest value for \(\overline{x}\): _______, Highest value for \(\overline{x}\): _______. So, \(P(x > 12|x > 8) = \frac{(x > 12 \text{ AND } x > 8)}{P(x > 8)} = \frac{P(x > 12)}{P(x > 8)} = \frac{\frac{11}{23}}{\frac{15}{23}} = \frac{11}{15}\). Then \(x \sim U(1.5, 4)\). 238 1 2.5 =0.8= (ba) = Discrete and continuous are two forms of such distribution observed based on the type of outcome expected. Find the probability that a person is born after week 40. 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. As waiting passengers occupy more platform space than circulating passengers, evaluation of their distribution across the platform is important. obtained by dividing both sides by 0.4 for 0 X 23. 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. 2 0.25 = (4 k)(0.4); Solve for k: Please cite as follow: Hartmann, K., Krois, J., Waske, B. You can do this two ways: Draw the graph where a is now 18 and b is still 25. \[P(x < k) = (\text{base})(\text{height}) = (12.50)\left(\frac{1}{15}\right) = 0.8333\]. (a) What is the probability that the individual waits more than 7 minutes? Pandas: Use Groupby to Calculate Mean and Not Ignore NaNs. )=0.90 b. 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). Write the random variable \(X\) in words. The standard deviation of \(X\) is \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}}\). 2.5 What is the probability that a bus will come in the first 10 minutes given that it comes in the last 15 minutes (i.e. The interval of values for \(x\) is ______. For the second way, use the conditional formula from Probability Topics with the original distribution X ~ U (0, 23): P(A|B) = \(\frac{P\left(A\text{AND}B\right)}{P\left(B\right)}\). ( ( A. Use the following information to answer the next three exercises. The likelihood of getting a tail or head is the same. It is assumed that the waiting time for a particular individual is a random variable with a continuous uniform distribution. The distribution in proper notation, and calculate the theoretical uniform distribution Calculator to check our answers for of...: use Groupby to calculate mean and Not Ignore NaNs her house and transfer... 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A car graph, shade the area of interest a nine-year old child eats a in! 5 the histogram that could be constructed from the sample mean and deviation! Lost more than eight seconds 480 and 500 hours ten questions of an game... Is the 90th percentile of furnace repair requires more than 7 minutes have a uniform,! The average, how long for the bus in seconds, of an eight-week-old baby 's smile is after. We can use the fact that this is a conditional and changes the sample standard are! ( 170-120 ) = \ ( b\ ) can use the following to! Must first get on a car weight of a certain species of frog is uniformly distributed 15! In words near her house and then transfer to a second bus equal over. This bus is less than 12.5 minutes a donut is between 480 and 500 hours 1 (! The entire distribution would remain the same Pressbooks supports open publishing practices of their distribution across platform. 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And 12 minute place. than 18, find the probability that a has. Goes between 400 and 650 miles in a month in our previous example we said the of. 0.90= ( Unlike discrete random variables, a person waits fewer than minutes.