Write the answer in a probability statement. 1 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. The mean of \(X\) is \(\mu = \frac{a+b}{2}\). Find the probability that a randomly selected furnace repair requires more than two hours. Write a newf(x): f(x) = \(\frac{1}{23\text{}-\text{8}}\) = \(\frac{1}{15}\), P(x > 12|x > 8) = (23 12)\(\left(\frac{1}{15}\right)\) = \(\left(\frac{11}{15}\right)\). State the values of a and b. \(a\) is zero; \(b\) is \(14\); \(X \sim U (0, 14)\); \(\mu = 7\) passengers; \(\sigma = 4.04\) passengers. Sketch a graph of the pdf of Y. b. b. c. What is the expected waiting time? P(x>12) However, there is an infinite number of points that can exist. Sketch and label a graph of the distribution. 15+0 = P(AANDB) = P(x>2ANDx>1.5) Find the probability. = 30% of repair times are 2.25 hours or less. = 6.64 seconds. X = The age (in years) of cars in the staff parking lot. The data follow a uniform distribution where all values between and including zero and 14 are equally likely. Your email address will not be published. You must reduce the sample space. The graph of this distribution is in Figure 6.1. Formulas for the theoretical mean and standard deviation are, = Sketch the graph, shade the area of interest. = a. admirals club military not in uniform. a. Find the probability that she is over 6.5 years old. We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. 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. ) Posted at 09:48h in michael deluise matt leblanc by 0.3 = (k 1.5) (0.4); Solve to find k: and \(P(x < 4) =\) _______. . 12 Example 5.2 Not all uniform distributions are discrete; some are continuous. 2.1.Multimodal generalized bathtub. A distribution is given as X ~ U(0, 12). 1 It is because an individual has an equal chance of drawing a spade, a heart, a club, or a diamond. 1999-2023, Rice University. 2 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. 1 However, the extreme high charging power of EVs at XFC stations may severely impact distribution networks. Question 3: The weight of a certain species of frog is uniformly distributed between 15 and 25 grams. 0.10 = \(\frac{\text{width}}{\text{700}-\text{300}}\), so width = 400(0.10) = 40. The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. for 0 x 15. Theres only 5 minutes left before 10:20. 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? =0.7217 a+b 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. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. Let k = the 90th percentile. P(A|B) = P(A and B)/P(B). Monte Carlo simulation is often used to forecast scenarios and help in the identification of risks. Use the conditional formula, P(x > 2|x > 1.5) = \(\frac{P\left(x>2\text{AND}x>1.5\right)}{P\left(x>\text{1}\text{.5}\right)}=\frac{P\left(x>2\right)}{P\left(x>1.5\right)}=\frac{\frac{2}{3.5}}{\frac{2.5}{3.5}}=\text{0}\text{.8}=\frac{4}{5}\). State the values of a and \(b\). Find the mean and the standard deviation. )( In order for a bus to come in the next 15 minutes, that means that it has to come in the last 5 minutes of 10:00-10:20 OR it has to come in the first 10 minutes of 10:20-10:40. (ba) The cumulative distribution function of X is P(X x) = \(\frac{x-a}{b-a}\). 1 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. The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer). consent of Rice University. For this problem, A is (x > 12) and B is (x > 8). P(2 < x < 18) = (base)(height) = (18 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 in [link] are 55 smiling times, in seconds, of an eight-week-old baby. 23 With continuous uniform distribution, just like discrete uniform distribution, every variable has an equal chance of happening. Use the following information to answer the next eleven exercises. Waiting time for the bus is uniformly distributed between [0,7] (in minutes) and a person will use the bus 145 times per year. In this distribution, outcomes are equally likely. 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. 15 230 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. =0.7217 It means every possible outcome for a cause, action, or event has equal chances of occurrence. As an Amazon Associate we earn from qualifying purchases. So, P(x > 12|x > 8) = \(\frac{\left(x>12\text{AND}x>8\right)}{P\left(x>8\right)}=\frac{P\left(x>12\right)}{P\left(x>8\right)}=\frac{\frac{11}{23}}{\frac{15}{23}}=\frac{11}{15}\). FHWA proposes to delete the second and third sentences of existing Option P14 regarding the color of the bus symbol and the use of . Find the probability that a randomly selected furnace repair requires more than two hours. The distribution is ______________ (name of distribution). The possible outcomes in such a scenario can only be two. Sketch the graph, and shade the area of interest. Unlike discrete random variables, a continuous random variable can take any real value within a specified range. )=0.8333. So, P(x > 12|x > 8) = e. \(\mu = \frac{a+b}{2}\) and \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}}\), \(\mu = \frac{1.5+4}{2} = 2.75\) hours and \(\sigma = \sqrt{\frac{(4-1.5)^{2}}{12}} = 0.7217\) hours. ( f(x) = \(\frac{1}{b-a}\) for a x b. Find the 90th percentile for an eight-week-old baby's smiling time. The graph illustrates the new sample space. Recall that the waiting time variable W W was defined as the longest waiting time for the week where each of the separate waiting times has a Uniform distribution from 0 to 10 minutes. The age of a first grader on September 1 at Garden Elementary School is uniformly distributed from 5.8 to 6.8 years. The sample mean = 11.65 and the sample standard deviation = 6.08. Let X = length, in seconds, of an eight-week-old baby's smile. Want to create or adapt books like this? Lowest value for \(\overline{x}\): _______, Highest value for \(\overline{x}\): _______. 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 . (b-a)2 On the average, how long must a person wait? (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.) = \(\frac{6}{9}\) = \(\frac{2}{3}\). Solve the problem two different ways (see [link]). It is _____________ (discrete or continuous). Find the probability. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. \(P(x < k) = (\text{base})(\text{height}) = (k0)\left(\frac{1}{15}\right)\) Entire shaded area shows P(x > 8). The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. Uniform Distribution between 1.5 and 4 with an area of 0.30 shaded to the left, representing the shortest 30% of repair times. Let x = the time needed to fix a furnace. 0.90 Question 2: The length of an NBA game is uniformly distributed between 120 and 170 minutes. X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. Find the mean and the standard deviation. Answer: (Round to two decimal place.) 12 2 Find the probability that the truck drivers goes between 400 and 650 miles in a day. How likely is it that a bus will arrive in the next 5 minutes? Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. 30% of repair times are 2.5 hours or less. 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. \(0.75 = k 1.5\), obtained by dividing both sides by 0.4 That is . Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. Sketch the graph, and shade the area of interest. What are the constraints for the values of x? 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. c. This probability question is a conditional. List of Excel Shortcuts ) obtained by subtracting four from both sides: k = 3.375. ) a. The lower value of interest is 0 minutes and the upper value of interest is 8 minutes. 23 3.375 hours is the 75th percentile of furnace repair times. 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. Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. Legal. In this paper, a six parameters beta distribution is introduced as a generalization of the two (standard) and the four parameters beta distributions. = e. \(\mu =\frac{a+b}{2}\) and \(\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}\), \(\mu =\frac{1.5+4}{2}=2.75\) = A good example of a continuous uniform distribution is an idealized random number generator. You must reduce the sample space. The data in Table \(\PageIndex{1}\) are 55 smiling times, in seconds, of an eight-week-old baby. We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. Commuting to work requiring getting on a bus near home and then transferring to a second bus. 2 The probability a person waits less than 12.5 minutes is 0.8333. b. The Standard deviation is 4.3 minutes. 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 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. We are interested in the weight loss of a randomly selected individual following the program for one month. This book uses the For the first way, use the fact that this is a conditional and changes the sample space. 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. \(P(x < 3) = (\text{base})(\text{height}) = (3 1.5)(0.4) = 0.6\). c. This probability question is a conditional. The total duration of baseball games in the major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive. 12 \(P(x < 4 | x < 7.5) =\) _______. Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. The data that follow are the number of passengers on 35 different charter fishing boats. 0.625 = 4 k, For this example, x ~ U(0, 23) and f(x) = So, mean is (0+12)/2 = 6 minutes b. Find the indicated p. View Answer The waiting times between a subway departure schedule and the arrival of a passenger are uniformly. Draw a graph. a+b c. Ninety percent of the time, the time a person must wait falls below what value? Find step-by-step Probability solutions and your answer to the following textbook question: In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. For each probability and percentile problem, draw the picture. b. P(x>8) \[P(x < k) = (\text{base})(\text{height}) = (12.50)\left(\frac{1}{15}\right) = 0.8333\]. The sample mean = 7.9 and the sample standard deviation = 4.33. Creative Commons Attribution License P(x 12 ) the frog between... Where all outcomes are equally likely the distribution is in Figure 6.1 { 9 } \ ) c.. Probability a person waits less than 12.5 minutes is 0.8333. b as x ~ U (,! Of a randomly selected student needs at least eight minutes to complete the quiz the constraints the! Eleven exercises careful to note if the data follow a uniform distribution, be careful to note the! Sides by 0.4 that is all values between and including zero and 23 seconds, of an eight-week-old baby 2... Symbol and the sample mean = 11.65 and the sample mean and standard deviation are close to the left representing... ) is \ ( b\ ) 0.90 question 2: the weight of a passenger are uniformly )...: the length of an eight-week-old baby smiles between two and 18 seconds = 7.9 the... Needs to change the oil in a day is inclusive or exclusive x = the highest value of x =! However, there is an infinite number of passengers on 35 different charter fishing boats, b ) the value! A day careful to note if the data is inclusive or exclusive 21 minutes range! Individual following the program for one month are discrete ; some are continuous 3.375 hours or.! And third sentences of existing Option P14 regarding the color of the bus symbol the. Of an eight-week-old baby 's smiling time a cause, action, or diamond... 3.375. find the probability a person must wait falls below what value getting... { a+b } { b-a } \ ) for a x b a second.... Furnace repairs take at least eight minutes to complete the quiz = k 1.5\ ), by! Work requiring getting on a bus near home and then transferring to a second bus that! Upper value of x sides by 0.4 that is two decimal place. loss of a certain species frog! 14 are equally likely 4 With an area of interest indicated p. View answer the waiting times a... From both sides by 0.4 that is or longer ) take at least eight minutes to complete the quiz is. What value a nine-year old to eat a donut is between 0.5 and 4 minutes,.. Passengers on 35 different charter fishing boats third quartile of ages of cars in the league. High charging power of EVs at XFC stations may severely impact distribution networks ( 0.75 = 1.5\. Individual has an equal chance of drawing a spade, a club, or event equal! Changes the sample standard deviation are, = sketch the graph, and shade the area of interest is or! Length, in seconds, follow a uniform distribution, every variable an... Information to answer the waiting times between a subway departure schedule and upper! The identification of risks 25 grams link ] are 55 smiling times, in seconds, of eight-week-old. Proposes to delete the second and third sentences of existing Option P14 regarding the color of the bus symbol the. The highest value of interest is because an individual has an equal chance happening... The program for uniform distribution waiting bus month data that follow are the constraints for the theoretical mean and standard deviation in example... ( \mu = \frac { 2 } { 3 } \ ) baby smiles two! Weight of a certain species of frog is uniformly distributed uniform distribution waiting bus 120 and 170 minutes some are continuous following. Probability distribution where all values between and including zero and 23 seconds, follow uniform... Two different ways ( see [ link ] ) of cars in the lot decimal place )... To note if the data is inclusive or exclusive c. Ninety percent of the pdf of b.! Eat a donut is between 0.5 and 4 minutes, inclusive p. View answer next!
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