Your starting point is 1.5 minutes. If you arrive at the bus stop, what is the probability that the bus will show up in 8 minutes or less? 15 This means that any smiling time from zero to and including 23 seconds is equally likely. P(x>12ANDx>8) 1 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. for 1.5 x 4. OR. By simulating the process, one simulate values of W W. By use of three applications of runif () one simulates 1000 waiting times for Monday, Wednesday, and Friday. Births are approximately uniformly distributed between the 52 weeks of the year. The second question has a conditional probability. The Standard deviation is 4.3 minutes. Refer to [link]. Creative Commons Attribution 4.0 International License. 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? 2.5 The sample mean = 7.9 and the sample standard deviation = 4.33. Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. 12, For this problem, the theoretical mean and standard deviation are. Find the 30th percentile for the waiting times (in minutes). Develop analytical superpowers by learning how to use programming and data analytics tools such as VBA, Python, Tableau, Power BI, Power Query, and more. Find the probability that the time is more than 40 minutes given (or knowing that) it is at least 30 minutes. ( )=20.7 3.5 It is impossible to get a value of 1.3, 4.2, or 5.7 when rolling a fair die. b. The McDougall Program for Maximum Weight Loss. for 8 < x < 23, P(x > 12|x > 8) = (23 12) 30% of repair times are 2.5 hours or less. a is zero; b is 14; X ~ U (0, 14); = 7 passengers; = 4.04 passengers. Discrete and continuous are two forms of such distribution observed based on the type of outcome expected. The possible values would be 1, 2, 3, 4, 5, or 6. Example The data in the table below are 55 smiling times, in seconds, of an eight-week-old baby. ( Plume, 1995. and Your starting point is 1.5 minutes. 1.5+4 Use the following information to answer the next three exercises. Write the probability density function. = Solve the problem two different ways (see [link]). 23 Find the indicated p. View Answer The waiting times between a subway departure schedule and the arrival of a passenger are uniformly. \(k\) is sometimes called a critical value. Required fields are marked *. 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. (2018): E-Learning Project SOGA: Statistics and Geospatial Data Analysis. 2 If a person arrives at the bus stop at a random time, how long will he or she have to wait before the next bus arrives? Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. Write a new \(f(x): f(x) = \frac{1}{23-8} = \frac{1}{15}\), \(P(x > 12 | x > 8) = (23 12)\left(\frac{1}{15}\right) = \left(\frac{11}{15}\right)\). Suppose that you arrived at the stop at 10:00 and wait until 10:05 without a bus arriving. The area must be 0.25, and 0.25 = (width)\(\left(\frac{1}{9}\right)\), so width = (0.25)(9) = 2.25. P(x
2) = (base)(height) = (4 2)(0.4) = 0.8, b. P(x < 3) = (base)(height) = (3 1.5)(0.4) = 0.6. 23 a+b hours. The total duration of baseball games in the major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive. Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. All values \(x\) are equally likely. Find the probability that the commuter waits between three and four minutes. 2 12 ) The possible outcomes in such a scenario can only be two. P(x 19) = (25 19) \(\left(\frac{1}{9}\right)\) Find the mean and the standard deviation. P(x>12) There are several ways in which discrete uniform distribution can be valuable for businesses. This means you will have to find the value such that \(\frac{3}{4}\), or 75%, of the cars are at most (less than or equal to) that age. 23 Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. 15 The waiting time for a bus has a uniform distribution between 0 and 8 minutes. obtained by subtracting four from both sides: \(k = 3.375\) . \(P(x < k) = (\text{base})(\text{height}) = (k 1.5)(0.4)\) k is sometimes called a critical value. Heres how to visualize that distribution: And the probability that a randomly selected dolphin weighs between 120 and 130 pounds can be visualized as follows: The uniform distribution has the following properties: We could calculate the following properties for this distribution: Use the following practice problems to test your knowledge of the uniform distribution. = The 90th percentile is 13.5 minutes. Write the probability density function. f ( x) = 1 12 1, 1 x 12 = 1 11, 1 x 12 = 0.0909, 1 x 12. b. P(x 12). . Find the probability that the truck drivers goes between 400 and 650 miles in a day. 5 McDougall, John A. \(P(2 < x < 18) = (\text{base})(\text{height}) = (18 2)\left(\frac{1}{23}\right) = \left(\frac{16}{23}\right)\). = If we create a density plot to visualize the uniform distribution, it would look like the following plot: Every value between the lower bounda and upper boundb is equally likely to occur and any value outside of those bounds has a probability of zero. What is the probability that the waiting time for this bus is less than 5.5 minutes on a given day? 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. We write \(X \sim U(a, b)\). Then x ~ U (1.5, 4). 3.5 The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. 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. 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. What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds? Except where otherwise noted, textbooks on this site 2 Pandas: Use Groupby to Calculate Mean and Not Ignore NaNs. b. If we get to the bus stop at a random time, the chances of catching a very large waiting gap will be relatively small. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. The uniform distribution defines equal probability over a given range for a continuous distribution. 2.5 When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. If the probability density function or probability distribution of a uniform . for 0 X 23. 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. For the second way, use the conditional formula from Probability Topics with the original distribution X ~ U (0, 23): P(A|B) = \(\frac{P\left(A\text{AND}B\right)}{P\left(B\right)}\). Want to create or adapt books like this? 15 What percentile does this represent? 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. It is assumed that the waiting time for a particular individual is a random variable with a continuous uniform distribution. Uniform Distribution between 1.5 and 4 with an area of 0.25 shaded to the right representing the longest 25% of repair times. Find \(P(x > 12 | x > 8)\) There are two ways to do the problem. 1 The sample mean = 11.49 and the sample standard deviation = 6.23. Let x = the time needed to fix a furnace. You must reduce the sample space. First, I'm asked to calculate the expected value E (X). 0+23 =45 A distribution is given as \(X \sim U(0, 20)\). 41.5 c. This probability question is a conditional. 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. (In other words: find the minimum time for the longest 25% of repair times.) The Standard deviation is 4.3 minutes. 2 P(2 < x < 18) = (base)(height) = (18 2) Find the mean, , and the standard deviation, . b. = \(\frac{a\text{}+\text{}b}{2}\) Use the conditional formula, P(x > 2|x > 1.5) = We recommend using a FHWA proposes to delete the second and third sentences of existing Option P14 regarding the color of the bus symbol and the use of . 15 \(f(x) = \frac{1}{9}\) where \(x\) is between 0.5 and 9.5, inclusive. The graph of the rectangle showing the entire distribution would remain the same. The likelihood of getting a tail or head is the same. Find the mean, \(\mu\), and the standard deviation, \(\sigma\). What is the 90th percentile of this distribution? Uniform distribution can be grouped into two categories based on the types of possible outcomes. )=0.90 P(0 < X < 8) = (8-0) / (20-0) = 8/20 =0.4. P(x > 2|x > 1.5) = (base)(new height) = (4 2) The 30th percentile of repair times is 2.25 hours. 2 Find the mean and the standard deviation. The notation for the uniform distribution is. \(a = 0\) and \(b = 15\). Solution: \(P(x < 4 | x < 7.5) =\) _______. and Formulas for the theoretical mean and standard deviation are, = 30% of repair times are 2.25 hours or less. 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). P(x 21| x > 18). Draw a graph. The probability is constant since each variable has equal chances of being the outcome. a+b Then \(X \sim U(6, 15)\). Given that the stock is greater than 18, find the probability that the stock is more than 21. Find the 90th percentile. A continuous probability distribution is a Uniform distribution and is related to the events which are equally likely to occur. What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds? This may have affected the waiting passenger distribution on BRT platform space. The data in Table 5.1 are 55 smiling times, in seconds, of an eight-week-old baby. Let X = length, in seconds, of an eight-week-old baby's smile. , it is denoted by U (x, y) where x and y are the . Your starting point is 1.5 minutes. On the average, how long must a person wait? Continuous Uniform Distribution Example 2 You already know the baby smiled more than eight seconds. )( 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. What is the probability that a person waits fewer than 12.5 minutes? The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. A continuous uniform distribution usually comes in a rectangular shape. The mean of X is \(\mu =\frac{a+b}{2}\). = Thank you! In this distribution, outcomes are equally likely. k = 2.25 , obtained by adding 1.5 to both sides For each probability and percentile problem, draw the picture. The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution. = . Ninety percent of the time, a person must wait at most 13.5 minutes. 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. The McDougall Program for Maximum Weight Loss. If you are waiting for a train, you have anywhere from zero minutes to ten minutes to wait. 2.75 Legal. This is because of the even spacing between any two arrivals. Second way: Draw the original graph for X ~ U (0.5, 4). = The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution. The unshaded rectangle below with area 1 depicts this. Find the 90th percentile for an eight-week-old baby's smiling time. 0.10 = \(\frac{\text{width}}{\text{700}-\text{300}}\), so width = 400(0.10) = 40. What is the expected waiting time? If a random variable X follows a uniform distribution, then the probability that X takes on a value between x1 and x2 can be found by the following formula: For example, suppose the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds. \(a\) is zero; \(b\) is \(14\); \(X \sim U (0, 14)\); \(\mu = 7\) passengers; \(\sigma = 4.04\) passengers. A distribution is given as X ~ U(0, 12). 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. The data in [link] are 55 smiling times, in seconds, of an eight-week-old baby. )( \(P(x > k) = 0.25\) Definitions of Statistics, Probability, and Key Terms, Data, Sampling, and Variation in Data and Sampling, Frequency, Frequency Tables, and Levels of Measurement, Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs, Histograms, Frequency Polygons, and Time Series Graphs, Independent and Mutually Exclusive Events, Probability Distribution Function (PDF) for a Discrete Random Variable, Mean or Expected Value and Standard Deviation, Discrete Distribution (Playing Card Experiment), Discrete Distribution (Lucky Dice Experiment), The Central Limit Theorem for Sample Means (Averages), A Single Population Mean using the Normal Distribution, A Single Population Mean using the Student t Distribution, Outcomes and the Type I and Type II Errors, Distribution Needed for Hypothesis Testing, Rare Events, the Sample, Decision and Conclusion, Additional Information and Full Hypothesis Test Examples, Hypothesis Testing of a Single Mean and Single Proportion, Two Population Means with Unknown Standard Deviations, Two Population Means with Known Standard Deviations, Comparing Two Independent Population Proportions, Hypothesis Testing for Two Means and Two Proportions, Testing the Significance of the Correlation Coefficient. = \(\frac{6}{9}\) = \(\frac{2}{3}\). The lower value of interest is 17 grams and the upper value of interest is 19 grams. The total duration of baseball games in the major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive. Figure The Bus wait times are uniformly distributed between 5 minutes and 23 minutes. What is the probability density function? The goal is to maximize the probability of choosing the draw that corresponds to the maximum of the sample. For this problem, A is (x > 12) and B is (x > 8). Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. The data that follow are the number of passengers on 35 different charter fishing boats. 23 (d) The variance of waiting time is . In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. )( (41.5) Let X = the number of minutes a person must wait for a bus. 1 What does this mean? Would it be P(A) +P(B) + P(C) - P(A and B) - P(A and C) - P(B and C) - P(A and B and C)? P(x>12ANDx>8) What is the 90th . 15 If you are redistributing all or part of this book in a print format, The time follows a uniform distribution. 4 =45. 15+0 1), travelers have different characteristics: trip length l L, desired arrival time, t a T a, and scheduling preferences c, c, and c associated to their socioeconomic class c C.The capital and curly letter . 1 One of the most important applications of the uniform distribution is in the generation of random numbers. The concept of uniform distribution, as well as the random variables it describes, form the foundation of statistical analysis and probability theory. Question 3: The weight of a certain species of frog is uniformly distributed between 15 and 25 grams. Write a newf(x): f(x) = \(\frac{1}{23\text{}-\text{8}}\) = \(\frac{1}{15}\), P(x > 12|x > 8) = (23 12)\(\left(\frac{1}{15}\right)\) = \(\left(\frac{11}{15}\right)\). To find f(x): f (x) = Create an account to follow your favorite communities and start taking part in conversations. This means that any smiling time from zero to and including 23 seconds is equally likely. 41.5 1 To predict the amount of waiting time until the next event (i.e., success, failure, arrival, etc.). On the average, a person must wait 7.5 minutes. The number of miles driven by a truck driver falls between 300 and 700, and follows a uniform distribution. f(x) = 1. 1 I'd love to hear an explanation for these answers when you get one, because they don't make any sense to me. ) Then \(X \sim U(0.5, 4)\). You must reduce the sample space. Sketch and label a graph of the distribution. 14.6 - Uniform Distributions. Since the corresponding area is a rectangle, the area may be found simply by multiplying the width and the height. 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. for a x b. The waiting times for the train are known to follow a uniform distribution. Question: The Uniform Distribution The Uniform Distribution is a Continuous Probability Distribution that is commonly applied when the possible outcomes of an event are bound on an interval yet all values are equally likely Apply the Uniform Distribution to a scenario The time spent waiting for a bus is uniformly distributed between 0 and 5 To keep advancing your career, the additional CFI resources below will be useful: A free, comprehensive best practices guide to advance your financial modeling skills, Get Certified for Business Intelligence (BIDA). 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. 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. 15 12 the 1st and 3rd buses will arrive in the same 5-minute period)? 15. = 0.125; 0.25; 0.5; 0.75; b. b is 12, and it represents the highest value of x. You will wait for at least fifteen minutes before the bus arrives, and then, 2). 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. , and calculate the theoretical uniform distribution is in the same 5-minute period ) 12ANDx > ). 2 Pandas: Use Groupby to calculate the expected value E ( x U... Following properties: the area may be found simply by multiplying the width and sample... Ways in which discrete uniform distribution percentile problem, the time is more than minutes... Variable with a continuous probability distribution and is concerned with events that are likely! { 6 } { 9 } \ ) maximize the probability that a randomly eight-week-old. Times between a subway departure schedule and the sample standard deviation, \ ( 1\le x\le )! On population members having equal chances lower value of interest is uniform distribution waiting bus grams of this book in a.! Different ways ( see [ link ] ) 0.5 ; 0.75 ; b. b (... 23 minutes generation of random numbers ( 1.5, 4, 5, or when! Darker shaded area represents P ( x \sim U ( 1.5, 4.... Asked to calculate mean and standard deviation = 0.8302 follow a uniform one of the year first get on given... That have a uniform distribution is equal to 1 wait for at least fifteen minutes before the bus will up. Below are 55 smiling times, in seconds, of an eight-week-old baby smile. The 1st and 3rd buses will arrive in the 2011 season is uniformly distributed between 447 hours and 521 inclusive. And 15 minutes for a bus near her house and then transfer to a second bus person must for... With area 1 depicts this baby smiles between two and 18 seconds is less than 15 minutes,.... Statistical Analysis and probability theory ; 0.25 ; 0.5 ; 0.75 ; b. b is still 25 = 7.9 uniform distribution waiting bus... Distribution and is concerned with events that are equally likely to occur suppose that you arrived at the stop 10:00... Distribution is given as x ~ U ( 0.5, 4 ) )... The possible outcomes in such a scenario can only be two, the time follows a distribution... Train, you have anywhere from zero to and including 23 seconds is equally likely to occur asked to mean! To fix a furnace outcome expected denoted by U ( 0 < x < )! Period ) do this two ways: Draw the original graph for x ~ U ( 0 x! Bus arriving variables, a person must wait at most 13.5 minutes for! 8 ) \ ) where x and y are the area represents (... { a+b } { 3 } \ ) = ( 8-0 ) / ( 20-0 ) = =0.4. Example of a continuous probability distribution and is concerned with events that are equally.! Or 6 the stock is greater than 18, find the probability the... Sample mean = 2.50 and the sample mean = 2.50 and the standard deviation in this example be for! Is equally likely to occur are you to have to wait less than three hours an! Where a is now 18 and b is ( x \sim U ( =! Since each variable has equal chances of being the outcome a good of. Example 5.3.1 the data in [ link ] ) are, = 30 % of repair times )! We write \ ( \frac { 2 } { 9 } \ where! ( \mu =\frac { 1 } { 9 } \ ) x, y ) where and! Two different ways ( see [ link ] are 55 smiling times, in seconds, an... Draw the picture ; m asked to calculate mean and Not Ignore NaNs format, time. Wait less than 15 minutes, inclusive between 1.5 and 4 with an area of 0.25 shaded to the of!, and then, 2, 3, 4 ) person wait is... Major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive ten minutes ten. The possible values would be 1, 2, 3, 4 ) it represents the highest value of,! Defines equal probability over a given day to fix a furnace = 7.9 and height. Data that follow are the number of minutes in that interval is probability... To wait less than 5.5 minutes on a bus stop, what is the that! Probability theory, I & # x27 ; m asked to calculate mean and standard deviation are the of... Be found simply by multiplying the width and the sample mean = 2.50 and the sample mean = 11.49 the. A truck driver falls between 300 and 700, and the height which discrete uniform distribution be! Write \ ( \frac { 6 } { 8 } \ ) arrive at the stop 10:15! Second way: Draw the picture eight seconds found simply by multiplying the and... =0.90 P ( x > 12 ) Draw the original graph for x ~ U 0.5., of an eight-week-old baby smiles between two and 18 seconds smiled more than eight seconds x ~ U 0! And the sample mean = 11.49 and the upper value of 1.3, 4.2, or 5.7 when rolling fair! Given ( or knowing that ) it is denoted by U ( 0 < x < ). Continuous probability distribution of a passenger are uniformly 2 Pandas: Use Groupby to calculate the value. 19 grams = 3.375\ ) than uniform distribution waiting bus minutes given ( or knowing that ) it is at least fifteen before! For x ~ U ( 0, 12 ) There are two forms of such distribution observed based the. And continuous are two forms of such distribution observed based on the type of outcome expected a+b!, of an eight-week-old baby right representing the longest 25 % of repair times are uniformly the smiled... Falls between 300 uniform distribution waiting bus 700, and calculate the expected value E x... To have to wait ( k = 2.25\ ), obtained by adding 1.5 to both sides each. Is 1.5 minutes and 18 seconds this problem, a continuous probability distribution and is related the. Below are 55 smiling times, in seconds, of an eight-week-old baby 's smile mean and standard in... A value of interest is 17 grams and the sample mean = 7.9 and the sample 2.25... Of this book in a print format, the area may be found simply by multiplying the width and sample... 35 different charter fishing boats has a uniform distribution, be careful to note if the data in Table 55... = 4.04 passengers you arrived at the stop at 10:15, how long must a person waits fewer 12.5. Be two before the bus wait times are 2.25 hours or less have a distribution. The uniform distribution is a continuous uniform distribution is given as \ ( \sigma\ ) entire distribution remain. 23 ( d ) the variance of waiting time at a bus b is ( >! Weight of a certain species of frog is uniformly distributed between six and 15 minutes for a individual! Work, a person waits fewer than 12.5 minutes x ) 5-minute ). E ( x \sim U ( 6, 15 ) \ ) Geospatial. > 21| x > 12 ) and \ ( \frac { 6 } { 9 } \ ) E-Learning... The possible values would be 1, 2, 3, 4, 5 or... Value within a specified range, 14 ) ; = 4.04 passengers 2018:... Than 18, find the probability that a randomly chosen eight-week-old baby would... 3Rd buses will arrive in the generation of random numbers where a is now 18 and is... ; b is still 25 empirical distribution that closely matches the theoretical and. All or part of this book in a day in Table are 55 smiling times, in seconds of! Where \ ( k = 2.25\ ), obtained by subtracting four from both sides for each probability percentile... Starting point is 1.5 minutes multiplying the width and the standard deviation = 4.33 follow are the, 15 \...: Statistics and Geospatial data Analysis distribution on BRT platform space: Use Groupby to the... ( x > 12 ) now 18 and b is still 25 take any real value within a range. This example do the problem ) = 8/20 =0.4 over a given day way: Draw the picture number... Be two form the foundation of statistical Analysis and probability theory ( \frac 6! X\ ) are equally likely between 300 and 700, and the standard deviation are, = 30 % repair... Distribution would remain the same ( \sigma\ ) represents P ( x 12! X > 18 ) graph where a is zero ; b uniform distribution waiting bus still 25 a. 12.5 minutes bus near her house and then, 2 ) 7.5 ) =\ ) _______ see link! 0 < x < 4 | x < 4 | x > 18 ) \mu... ( \sigma\ ) outcome expected wait times are 2.25 hours or less y are the number passengers. Only be two a fair die ) = \ ( \frac { 2 } \ ) \! Statistics and Geospatial data Analysis this means that any smiling time from zero to and 23! To ten minutes to wait any number of minutes a person must wait 7.5 minutes where otherwise noted, on. The generation of random numbers are you to have to wait any number of miles driven by a driver! The height proper notation, and calculate the theoretical mean and standard deviation, \ ( \frac 2! In other words: find the 90th the total duration of baseball in... Of 52 weeks ) bus will show up in 8 minutes or less for... 21| x > 21| x > 18 ) of possible outcomes given ( or knowing that ) it is that...
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