1.5+4 Solution 1: The minimum amount of time youd have to wait is 0 minutes and the maximum amount is 20 minutes. What is the probability that a person waits fewer than 12.5 minutes? Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . = What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds? A uniform distribution is a type of symmetric probability distribution in which all the outcomes have an equal likelihood of occurrence. Let X= the number of minutes a person must wait for a bus. A good example of a discrete uniform distribution would be the possible outcomes of rolling a 6-sided die. 2.1.Multimodal generalized bathtub. 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. What does this mean? Find the third quartile of ages of cars in the lot. The sample mean = 7.9 and the sample standard deviation = 4.33. b. Ninety percent of the smiling times fall below the 90th percentile, k, so P(x < k) = 0.90. ) There are several ways in which discrete uniform distribution can be valuable for businesses. admirals club military not in uniform. It is defined by two parameters, x and y, where x = minimum value and y = maximum value. For example, it can arise in inventory management in the study of the frequency of inventory sales. P(2 < x < 18) = (base)(height) = (18 2) So, P(x > 21|x > 18) = (25 21)\(\left(\frac{1}{7}\right)\) = 4/7. The data that follow record the total weight, to the nearest pound, of fish caught by passengers on 35 different charter fishing boats on one summer day. k = 2.25 , obtained by adding 1.5 to both sides 2 Possible waiting times are along the horizontal axis, and the vertical axis represents the probability. Random sampling because that method depends on population members having equal chances. obtained by dividing both sides by 0.4 1 Then x ~ U (1.5, 4). = 233K views 3 years ago This statistics video provides a basic introduction into continuous probability distribution with a focus on solving uniform distribution problems. Post all of your math-learning resources here. The waiting times for the train are known to follow a uniform distribution. On the average, a person must wait 7.5 minutes. Suppose that the value of a stock varies each day from 16 to 25 with a uniform distribution. What is the height of f(x) for the continuous probability distribution? Thank you! 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 amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. a = smallest X; b = largest X, The standard deviation is \(\sigma =\sqrt{\frac{{\left(b\text{}a\right)}^{2}}{12}}\), Probability density function:\(f\left(x\right)=\frac{1}{b-a}\) for \(a\le X\le b\), 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) = (base)(height) = (d c)\(\left(\frac{1}{b-a}\right)\). hours and The waiting times for the train are known to follow a uniform distribution. 2.5 ) P(x>1.5) Find the third quartile of ages of cars in the lot. Second way: Draw the original graph for X ~ U (0.5, 4). 1 \(P(x > 2|x > 1.5) = (\text{base})(\text{new height}) = (4 2)(25)\left(\frac{2}{5}\right) =\) ? If the waiting time (in minutes) at each stop has a uniform distribution with A = 0 and B = 5, then it can be shown that the total waiting time Y has the pdf f(y) = 1 25 y 0 y < 5 2 5 1 25 y 5 y 10 0 y < 0 or y > 10 P(x>12ANDx>8) 4 FHWA proposes to delete the second and third sentences of existing Option P14 regarding the color of the bus symbol and the use of . For example, in our previous example we said the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds. = The waiting time for a bus has a uniform distribution between 0 and 10 minutes. (b) The probability that the rider waits 8 minutes or less. Develop analytical superpowers by learning how to use programming and data analytics tools such as VBA, Python, Tableau, Power BI, Power Query, and more. The cumulative distribution function of X is P(X x) = \(\frac{x-a}{b-a}\). The data in (Figure) are 55 smiling times, in seconds, of an eight-week-old baby. The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer). 23 (ba) What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? 2 The probability density function is \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\). What is P(2 < x < 18)? Answer: (Round to two decimal place.) 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). (41.5) Sketch and label a graph of the distribution. Find the probability that the truck drivers goes between 400 and 650 miles in a day. Let X = the time, in minutes, it takes a nine-year old child to eat a donut. b. Can you take it from here? Find the probability that the time is between 30 and 40 minutes. P(X > 19) = (25 19) \(\left(\frac{1}{9}\right)\) The Standard deviation is 4.3 minutes. 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. Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. c. Find the 90th percentile. ( Refer to Example 5.2. Lowest value for \(\overline{x}\): _______, Highest value for \(\overline{x}\): _______. \(f(x) = \frac{1}{9}\) where \(x\) is between 0.5 and 9.5, inclusive. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. We are interested in the length of time a commuter must wait for a train to arrive. \(a\) is zero; \(b\) is \(14\); \(X \sim U (0, 14)\); \(\mu = 7\) passengers; \(\sigma = 4.04\) passengers. a. \(X\) is continuous. 15 = P(x > 2|x > 1.5) = (base)(new height) = (4 2) The number of miles driven by a truck driver falls between 300 and 700, and follows a uniform distribution. Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}} = \sqrt{\frac{(12-0)^{2}}{12}} = 4.3\). As the question stands, if 2 buses arrive, that is fine, because at least 1 bus arriving is satisfied. What is the probability that the waiting time for this bus is less than 6 minutes on a given day? View full document See Page 1 1 / 1 point c. This probability question is a conditional. b is 12, and it represents the highest value of x. You already know the baby smiled more than eight seconds. k=(0.90)(15)=13.5 f(x) = For example, we want to predict the following: The amount of timeuntilthe customer finishes browsing and actually purchases something in your store (success). 41.5 Write the probability density function. 23 At least how many miles does the truck driver travel on the furthest 10% of days? A continuous uniform distribution usually comes in a rectangular shape. 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. \(P(x > k) = 0.25\) The Manual on Uniform Traffic Control Devices for Streets and Highways (MUTCD) is incorporated in FHWA regulations and recognized as the national standard for traffic control devices used on all public roads. a. Write the answer in a probability statement. 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. . Legal. The sample mean = 11.65 and the sample standard deviation = 6.08. 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. Find the probability that the truck driver goes more than 650 miles in a day. Sketch the graph, and shade the area of interest. The Structured Query Language (SQL) comprises several different data types that allow it to store different types of information What is Structured Query Language (SQL)? The distribution is ______________ (name of distribution). It is assumed that the waiting time for a particular individual is a random variable with a continuous uniform distribution. 15 1 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. c. Ninety percent of the time, the time a person must wait falls below what value? For this example, x ~ U(0, 23) and f(x) = 1 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. 1. For the first way, use the fact that this is a conditional and changes the sample space. Use the conditional formula, P(x > 2|x > 1.5) = 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. 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. 12 So, mean is (0+12)/2 = 6 minutes b. 0.10 = \(\frac{\text{width}}{\text{700}-\text{300}}\), so width = 400(0.10) = 40. 2 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 data follow a uniform distribution where all values between and including zero and 14 are equally likely. Define the random . 23 Write a new \(f(x): f(x) = \frac{1}{23-8} = \frac{1}{15}\), \(P(x > 12 | x > 8) = (23 12)\left(\frac{1}{15}\right) = \left(\frac{11}{15}\right)\). Press question mark to learn the rest of the keyboard shortcuts. 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