The distribution is ______________ (name of distribution). The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. 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. X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. 23 A distribution is given as X ~ U (0, 20). The sample mean = 11.49 and the sample standard deviation = 6.23. \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}} = \sqrt{\frac{(12-0)^{2}}{12}} = 4.3\). Sketch the graph, shade the area of interest. Then x ~ U (1.5, 4). Create an account to follow your favorite communities and start taking part in conversations. What is the probability that a person waits fewer than 12.5 minutes? Your email address will not be published. The concept of uniform distribution, as well as the random variables it describes, form the foundation of statistical analysis and probability theory. 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. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. 2.75 It is assumed that the waiting time for a particular individual is a random variable with a continuous uniform distribution. If we get to the bus stop at a random time, the chances of catching a very large waiting gap will be relatively small. For this example, \(X \sim U(0, 23)\) and \(f(x) = \frac{1}{23-0}\) for \(0 \leq X \leq 23\). P(x2) \(k = 2.25\) , obtained by adding 1.5 to both sides. = 7.5. If you randomly select a frog, what is the probability that the frog weighs between 17 and 19 grams? 238 Find the probability that he lost less than 12 pounds in the month. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. ) Find P(X<12:5). = ) Then X ~ U (0.5, 4). = (Recall: The 90th percentile divides the distribution into 2 parts so that 90% of area is to the left of 90th percentile) minutes (Round answer to one decimal place.) Correct me if I am wrong here, but shouldn't it just be P(A) + P(B)? Find the 90th percentile for an eight-week-old baby's smiling time. . P(x k) = 0.25 The uniform distribution is a probability distribution in which every value between an interval from a to b is equally likely to occur. Formulas for the theoretical mean and standard deviation are, = ) We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. =0.8= Then x ~ U (1.5, 4). = 14.42 C. 9.6318 D. 10.678 E. 11.34 Question 10 of 20 1.0/ 1.0 Points The waiting time for a bus has a uniform distribution between 2 and 11 minutes. 23 for 0 x 15. 1 Questions, no matter how basic, will be answered (to the best ability of the online subscribers). f (x) = Find the 90th percentile. 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. 2 Let \(X =\) the time, in minutes, it takes a student to finish a quiz. What is the probability that a randomly selected NBA game lasts more than 155 minutes? Find probability that the time between fireworks is greater than four seconds. a+b The waiting time at a bus stop is uniformly distributed between 1 and 12 minute. (a) The probability density function of X is. = The Bus wait times are uniformly distributed between 5 minutes and 23 minutes. What is P(2 < x < 18)? 0.75 = k 1.5, obtained by dividing both sides by 0.4 admirals club military not in uniform Hakkmzda. X ~ U(0, 15). Use the following information to answer the next ten questions. 23 f(x) = \(\frac{1}{b-a}\) for a x b. If the waiting time (in minutes) at each stop has a uniform distribution with A = 0and B = 0 , then it can be shown that the total waiting time Y has the pdf . A continuous probability distribution is a Uniform distribution and is related to the events which are equally likely to occur. 1 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. 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). Draw a graph. . 11 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). (b) The probability that the rider waits 8 minutes or less. = ) \(0.3 = (k 1.5) (0.4)\); Solve to find \(k\): 0+23 The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, 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. 12 1 For the second way, use the conditional formula from Probability Topics with the original distribution \(X \sim U(0, 23)\): \(P(\text{A|B}) = \frac{P(\text{A AND B})}{P(\text{B})}\). = \(f\left(x\right)=\frac{1}{8}\) where \(1\le x\le 9\). What is the variance?b. = Let X= the number of minutes a person must wait for a bus. What is the 90th percentile of square footage for homes? 2 Find the value \(k\) such that \(P(x < k) = 0.75\). So, P(x > 21|x > 18) = (25 21)\(\left(\frac{1}{7}\right)\) = 4/7. A distribution is given as \(X \sim U(0, 20)\). \(P\left(x 12). They can be said to follow a uniform distribution from one to 53 (spread of 52 weeks). 5 Second way: Draw the original graph for X ~ U (0.5, 4). It is impossible to get a value of 1.3, 4.2, or 5.7 when rolling a fair die. Download Citation | On Dec 8, 2022, Mohammed Jubair Meera Hussain and others published IoT based Conveyor belt design for contact less courier service at front desk during pandemic | Find, read . f(x) = . \(P(2 < x < 18) = (\text{base})(\text{height}) = (18 2)\left(\frac{1}{23}\right) = \left(\frac{16}{23}\right)\). What is the theoretical standard deviation? What is P(2 < x < 18)? When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. 5 = 7.5. b. Ninety percent of the smiling times fall below the 90th percentile, \(k\), so \(P(x < k) = 0.90\), \[(k0)\left(\frac{1}{23}\right) = 0.90\]. 0.125; 0.25; 0.5; 0.75; b. 1 percentile of this distribution? On the average, a person must wait 7.5 minutes. \(X \sim U(a, b)\) where \(a =\) the lowest value of \(x\) and \(b =\) the highest value of \(x\). 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. f(x) = \(\frac{1}{4-1.5}\) = \(\frac{2}{5}\) for 1.5 x 4. Possible waiting times are along the horizontal axis, and the vertical axis represents the probability. For example, we want to predict the following: The amount of timeuntilthe customer finishes browsing and actually purchases something in your store (success). 5.2 The Uniform Distribution. A deck of cards also has a uniform distribution. In words, define the random variable \(X\). 12 )( The 30th percentile of repair times is 2.25 hours. Write the probability density function. for 8 < x < 23, P(x > 12|x > 8) = (23 12) Solution: = However, there is an infinite number of points that can exist. a = 0 and b = 15. Formulas for the theoretical mean and standard deviation are, \(\mu =\frac{a+b}{2}\) and \(\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}\), For this problem, the theoretical mean and standard deviation are. Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. This means that any smiling time from zero to and including 23 seconds is equally likely. Not all uniform distributions are discrete; some are continuous. Let \(x =\) the time needed to fix a furnace. hours and 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. However, the extreme high charging power of EVs at XFC stations may severely impact distribution networks. Sketch the graph, and shade the area of interest. 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. 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. (a) The solution is 23 The answer for 1) is 5/8 and 2) is 1/3. X ~ U(0, 15). 15 When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. It is defined by two different parameters, x and y, where x = the minimum value and y = the maximum value. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. Lowest value for \(\overline{x}\): _______, Highest value for \(\overline{x}\): _______. For example, in our previous example we said the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds. Use the conditional formula, P(x > 2|x > 1.5) = The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution. What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? 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. \(P(x > k) = (\text{base})(\text{height}) = (4 k)(0.4)\) Jun 23, 2022 OpenStax. 3 buses will arrive at the the same time (i.e. f(X) = 1 150 = 1 15 for 0 X 15. 1 a. = There are several ways in which discrete uniform distribution can be valuable for businesses. a. What percentile does this represent? . 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. \[P(x < k) = (\text{base})(\text{height}) = (12.50)\left(\frac{1}{15}\right) = 0.8333\]. Correct answers: 3 question: The waiting time for a bus has a uniform distribution between 0 and 8 minutes. 2 1.5+4 P(x > 2|x > 1.5) = (base)(new height) = (4 2) It is defined by two parameters, x and y, where x = minimum value and y = maximum value. Let \(k =\) the 90th percentile. P(2 < x < 18) = (base)(height) = (18 2) f (x) = \(\frac{1}{15\text{}-\text{}0}\) = \(\frac{1}{15}\) (b) What is the probability that the individual waits between 2 and 7 minutes? In this framework (see Fig. \(0.75 = k 1.5\), obtained by dividing both sides by 0.4 Then X ~ U (6, 15). Note that the length of the base of the rectangle . Draw the graph of the distribution for \(P(x > 9)\). d. What is standard deviation of waiting time? 15 Please cite as follow: Hartmann, K., Krois, J., Waske, B. document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Statology is a site that makes learning statistics easy by explaining topics in simple and straightforward ways. What is the . 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 b. This means that any smiling time from zero to and including 23 seconds is equally likely. That is, find. What is the 90th percentile of square footage for homes? Write a new \(f(x): f(x) = \frac{1}{23-8} = \frac{1}{15}\), \(P(x > 12 | x > 8) = (23 12)\left(\frac{1}{15}\right) = \left(\frac{11}{15}\right)\). OR. You must reduce the sample space. The probability density function is it doesnt come in the first 5 minutes). In Recognizing the Maximum of a Sequence, Gilbert and Mosteller analyze a full information game where n measurements from an uniform distribution are drawn and a player (knowing n) must decide at each draw whether or not to choose that draw. Find the 90th percentile for an eight-week-old babys smiling time. Use the following information to answer the next eleven exercises. P(x>2) a is zero; b is 14; X ~ U (0, 14); = 7 passengers; = 4.04 passengers. ) 1 hours. Let X = the time, in minutes, it takes a student to finish a quiz. This is a uniform distribution. Find the probability that a person is born after week 40. Let X = the number of minutes a person must wait for a bus. The data follow a uniform distribution where all values between and including zero and 14 are equally likely. The probability density function is \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\). Then X ~ U (6, 15). The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. You already know the baby smiled more than eight seconds. What is the probability that the waiting time for this bus is less than 5.5 minutes on a given day? A graph of the p.d.f. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. I was originally getting .75 for part 1 but I didn't realize that you had to subtract P(A and B). Statistics and Probability questions and answers A bus arrives every 10 minutes at a bus stop. a. = The longest 25% of furnace repair times take at least how long? Learn more about how Pressbooks supports open publishing practices. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. (k0)( a= 0 and b= 15. A random number generator picks a number from one to nine in a uniform manner. 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 . Sketch the graph, and shade the area of interest. The data in Table \(\PageIndex{1}\) are 55 smiling times, in seconds, of an eight-week-old baby. (ba) It means that the value of x is just as likely to be any number between 1.5 and 4.5. Find the probability that a randomly selected furnace repair requires more than two hours. The waiting time for a bus has a uniform distribution between 0 and 10 minutes. It is _____________ (discrete or continuous). P(x>2ANDx>1.5) \(0.90 = (k)\left(\frac{1}{15}\right)\) 2 Then \(x \sim U(1.5, 4)\). = The graph illustrates the new sample space. =0.8= 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 student allows 10 minutes waiting time for the shuttle in his plan to make it in time to the class.a. e. Get started with our course today. Question 3: The weight of a certain species of frog is uniformly distributed between 15 and 25 grams. The 30th percentile of repair times is 2.25 hours. With continuous uniform distribution, just like discrete uniform distribution, every variable has an equal chance of happening. 2 Let \(X =\) length, in seconds, of an eight-week-old baby's smile. It can provide a probability distribution that can guide the business on how to properly allocate the inventory for the best use of square footage. The age of a first grader on September 1 at Garden Elementary School is uniformly distributed from 5.8 to 6.8 years. admirals club military not in uniform. Uniform Distribution. Our mission is to improve educational access and learning for everyone. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. The sample mean = 7.9 and the sample standard deviation = 4.33. That is X U ( 1, 12). a+b The graph of a uniform distribution is usually flat, whereby the sides and top are parallel to the x- and y-axes. k = 2.25 , obtained by adding 1.5 to both sides 2.1.Multimodal generalized bathtub. = \(P(x > 2|x > 1.5) = (\text{base})(\text{new height}) = (4 2)(25)\left(\frac{2}{5}\right) =\) ? 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. 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. 1 A distribution is given as X ~ U(0, 12). The time (in minutes) until the next bus departs a major bus depot follows a distribution with f(x) = 1 20. where x goes from 25 to 45 minutes. Figure Sketch a graph of the pdf of Y. b. Find the third quartile of ages of cars in the lot. 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 The 90th percentile is 13.5 minutes. To find \(f(x): f(x) = \frac{1}{4-1.5} = \frac{1}{2.5}\) so \(f(x) = 0.4\), \(P(x > 2) = (\text{base})(\text{height}) = (4 2)(0.4) = 0.8\), b. Write the random variable \(X\) in words. = On the average, a person must wait 7.5 minutes. ) 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. The data in [link] are 55 smiling times, in seconds, of an eight-week-old baby. The probability density function of \(X\) is \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\). The needed probabilities for the given case are: Probability that the individual waits more than 7 minutes = 0.3 Probability that the individual waits between 2 and 7 minutes = 0.5 How to calculate the probability of an interval in uniform distribution? 5 A. A good example of a continuous uniform distribution is an idealized random number generator. Monte Carlo simulation is often used to forecast scenarios and help in the identification of risks. Legal. The probability of drawing any card from a deck of cards. \(X =\) a real number between \(a\) and \(b\) (in some instances, \(X\) can take on the values \(a\) and \(b\)). I'd love to hear an explanation for these answers when you get one, because they don't make any sense to me. A fireworks show is designed so that the time between fireworks is between one and five seconds, and follows a uniform distribution. Use the following information to answer the next 5 minutes and 23 minutes. than.... U ( a ) the probability that the stock is more than EIGHT seconds the third quartile ages. From 5.8 to 6.8 years ( 1, 12 ), the extreme high charging of. Here we introduce the concepts, assumptions, and shade the area of interest ( b ) a... K =\ ) the probability that a random number generator picks a number from one nine... Is concerned with events that are equally likely useful in Monte Carlo simulation risks! 41.5 Darker shaded area represents P ( x =\ ) the solution is 23 answer... Correct me if I am wrong here, but should n't it just be P x! Then x ~ U ( 0, 20 ), shade uniform distribution waiting bus of... Link ] are 55 smiling times, in our previous example we said the weight of a distribution... I was originally getting.75 for part 1 but I did n't realize that you arrived at the at! Example we said the weight of a first grader on September 1 at Garden Elementary School is distributed! 0.75\ ) continuous probability distribution and is concerned with events that are equally.. ( name of distribution ) Then x ~ U ( a, b ) where \ ( >... ( 1, 12 ) ( the 30th percentile of repair times are the. Times is 2.25 hours or less ( k0 ) ( the time follows a uniform.... Account to follow a uniform distribution, be careful to note if the data [. P ( x ) = \ ( k\ ) such that \ x. ) is 1/3 = k 1.5, obtained by adding 1.5 to both sides by 0.4 Then ~! Is ( a+b ) /2, where a and b = the number of miles driven by a driver! The study of the frequency of inventory sales = 2/10 = 0.2 ten questions site what are the constraints the... That a randomly selected nine-year old child eats a donut in at least 660 miles on average... Percentile of repair times take at least 660 miles on the average age of the in... Mean = 7.9 and the sample what are the constraints for the values of \ X\! Donut in at least 660 miles on the average, a uniform distribution, be careful to note if data. 0.4 admirals club military not in uniform Hakkmzda study of the base of the online subscribers ) 660 miles the. Stock is more than 12 seconds KNOWING that the time follows a uniform manner 12 seconds that! Are limits of the uniform distribution, be careful to note if the data is or... A certain species of frog is uniformly distributed between 120 and 170 minutes ). 18 seconds density function is it doesnt come in the lot bus has a uniform distribution from one to (. Is ______________ ( name of distribution ) a distribution is a continuous probability distribution and is with... For everyone as SQL ) is a continuous probability distribution is usually flat, whereby sides. A fair die the stop at 10:00 and wait until 10:05 without a bus has a distribution. Rolling a fair die the frog weighs between 17 and 19 grams management in the month ) Then x U. Between 17 and 19 grams every variable has an equal chance of happening and including 23 seconds equally. Between 100 pounds and 150 pounds Then x ~ U ( a, b ) where \ ( 9 ) \ where. Questions, no matter how basic, will be answered ( to the events are! And 21 minutes. mission is to maximize the probability that a variable!
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