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. 23 . The total duration of baseball games in the major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive. Example The data in the table below are 55 smiling times, in seconds, of an eight-week-old baby. On the average, a person must wait 7.5 minutes. Draw a graph. It means that the value of x is just as likely to be any number between 1.5 and 4.5. 1 (a) The solution is Excel shortcuts[citation CFIs free Financial Modeling Guidelines is a thorough and complete resource covering model design, model building blocks, and common tips, tricks, and What are SQL Data Types? Since 700 40 = 660, the drivers travel at least 660 miles on the furthest 10% of days. = 6.64 seconds. Example 5.2 \(P(x < k) = 0.30\) P(AANDB) 1.5+4 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 \(\frac{4}{5}\). Write the random variable \(X\) in words. P(155 < X < 170) = (170-155) / (170-120) = 15/50 = 0.3. 41.5 What is the probability density function? When working out problems that have a uniform distribution, be careful to note if the data are inclusive or exclusive of endpoints. Uniform Distribution between 1.5 and 4 with an area of 0.25 shaded to the right representing the longest 25% of repair times. Uniform distribution: happens when each of the values within an interval are equally likely to occur, so each value has the exact same probability as the others over the entire interval givenA Uniform distribution may also be referred to as a Rectangular distribution 15 Find the probability that a randomly chosen car in the lot was less than four years old. 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. The 90th percentile is 13.5 minutes. P(x>1.5) Sketch the graph of the probability distribution. Draw the graph of the distribution for P(x > 9). 1 14.6 - Uniform Distributions. On the average, how long must a person wait? 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. P(x>1.5) =0.8= \(\mu = \frac{a+b}{2} = \frac{15+0}{2} = 7.5\). a+b Therefore, the finite value is 2. We are interested in the length of time a commuter must wait for a train to arrive. They can be said to follow a uniform distribution from one to 53 (spread of 52 weeks). P(17 < X < 19) = (19-17) / (25-15) = 2/10 = 0.2. 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: . The time follows a uniform distribution. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. Find the mean and the standard deviation. ) \(a =\) 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 \(a < x < b\). For the first way, use the fact that this is a conditional and changes the sample space. ) (b-a)2 The second question has a conditional probability. then you must include on every digital page view the following attribution: Use the information below to generate a citation. We write \(X \sim U(a, b)\). Random sampling because that method depends on population members having equal chances. P(x>12) 2 State the values of a and \(b\). 2.5 2 41.5 View full document See Page 1 1 / 1 point The 30th percentile of repair times is 2.25 hours. = Unlike discrete random variables, a continuous random variable can take any real value within a specified range. 2.75 What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds? When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. Use the following information to answer the next ten questions. The number of miles driven by a truck driver falls between 300 and 700, and follows a uniform distribution. The mean of uniform distribution is (a+b)/2, where a and b are limits of the uniform distribution. = 7.5. 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. 2 c. Find the 90th percentile. (b-a)2 Find the probability that the value of the stock is more than 19. If you are waiting for a train, you have anywhere from zero minutes to ten minutes to wait. Sketch and label a graph of the distribution. 15 Uniform Distribution. \(P(x < k) = (\text{base})(\text{height}) = (k0)\left(\frac{1}{15}\right)\) A continuous uniform distribution is a statistical distribution with an infinite number of equally likely measurable values. Please cite as follow: Hartmann, K., Krois, J., Waske, B. 12 The data in Table \(\PageIndex{1}\) are 55 smiling times, in seconds, of an eight-week-old baby. So, P(x > 21|x > 18) = (25 21)\(\left(\frac{1}{7}\right)\) = 4/7. a = 0 and b = 15. )=0.8333 Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. What is the theoretical standard deviation? What is \(P(2 < x < 18)\)? Sixty percent of commuters wait more than how long for the train? , it is denoted by U (x, y) where x and y are the . The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. What is the probability that a randomly selected NBA game lasts more than 155 minutes? for 0 x 15. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. It is generally denoted by u (x, y). 15 15 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)=\left\{\begin{array}{cc} \frac . That is X U ( 1, 12). Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. a person has waited more than four minutes is? What is the probability that the rider waits 8 minutes or less? 2 (b) The probability that the rider waits 8 minutes or less. Can you take it from here? That is, find. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. I'd love to hear an explanation for these answers when you get one, because they don't make any sense to me. What is P(2 < x < 18)? The data in [link] are 55 smiling times, in seconds, of an eight-week-old baby. The Bus wait times are uniformly distributed between 5 minutes and 23 minutes. Find the mean, \(\mu\), and the standard deviation, \(\sigma\). What is the 90th percentile of square footage for homes? On the average, how long must a person wait? P(x 12 | x > 8) = (23 12)\left(\frac{1}{15}\right) = \left(\frac{11}{15}\right)\). The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. = Theres only 5 minutes left before 10:20. a. a. (In other words: find the minimum time for the longest 25% of repair times.) Extreme fast charging (XFC) for electric vehicles (EVs) has emerged recently because of the short charging period. 23 Find the probability that the time is at most 30 minutes. In reality, of course, a uniform distribution is . There are several ways in which discrete uniform distribution can be valuable for businesses. Best Buddies Turkey Ekibi; Videolar; Bize Ulan; admirals club military not in uniform 27 ub. 15+0 0.25 = (4 k)(0.4); Solve for k: For this problem, \(\text{A}\) is (\(x > 12\)) and \(\text{B}\) is (\(x > 8\)). = Let X = the time needed to change the oil on a car. The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution. Refer to Example 5.3.1. This means that any smiling time from zero to and including 23 seconds is equally likely. Find the average age of the cars in the lot. A good example of a continuous uniform distribution is an idealized random number generator. 15 Not all uniform distributions are discrete; some are continuous. All values \(x\) are equally likely. 12 Find the 90th percentile for an eight-week-old baby's smiling time. X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. \(f(x) = \frac{1}{9}\) where \(x\) is between 0.5 and 9.5, inclusive. A random number generator picks a number from one to nine in a uniform manner. What does this mean? Answer: (Round to two decimal places.) 3.5 2 The waiting time for a bus has a uniform distribution between 2 and 11 minutes. P(x>8) That is . The percentage of the probability is 1 divided by the total number of outcomes (number of passersby). Use the following information to answer the next eleven exercises. Find the 90thpercentile. P(x>2ANDx>1.5) If \(X\) has a uniform distribution where \(a < x < b\) or \(a \leq x \leq b\), then \(X\) takes on values between \(a\) and \(b\) (may include \(a\) and \(b\)). obtained by dividing both sides by 0.4 A continuous uniform distribution usually comes in a rectangular shape. =0.8= Uniform distribution is the simplest statistical distribution. This is a modeling technique that uses programmed technology to identify the probabilities of different outcomes. = 5 So, P(x > 12|x > 8) = 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. percentile of this distribution? You already know the baby smiled more than eight seconds. To me I thought I would just take the integral of 1/60 dx from 15 to 30, but that is not correct. 233K views 3 years ago This statistics video provides a basic introduction into continuous probability distribution with a focus on solving uniform distribution problems. The distribution can be written as X ~ U(1.5, 4.5). a. 2 Then x ~ U (1.5, 4). 1 For the first way, use the fact that this is a conditional and changes the sample space. (d) The variance of waiting time is . Not sure how to approach this problem. The probability density function is Draw a graph. If you arrive at the stop at 10:15, how likely are you to have to wait less than 15 minutes for a bus? 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. In this framework (see Fig. The concept of uniform distribution, as well as the random variables it describes, form the foundation of statistical analysis and probability theory. 15 What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? b. Ninety percent of the smiling times fall below the 90th percentile, k, so P(x < k) = 0.90. 23 = 11.50 seconds and = Let \(X =\) the time, in minutes, it takes a nine-year old child to eat a donut. What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds? It is _____________ (discrete or continuous). Find P(x > 12|x > 8) There are two ways to do the problem. Let x = the time needed to fix a furnace. The answer for 1) is 5/8 and 2) is 1/3. c. Ninety percent of the time, the time a person must wait falls below what value? 15 What is the 90th percentile of this distribution? Draw a graph. (In other words: find the minimum time for the longest 25% of repair times.) Find the probability that a different 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. Your probability of having to wait any number of minutes in that interval is the same. 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. 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. a+b The data follow a uniform distribution where all values between and including zero and 14 are equally likely. Find the indicated p. 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