The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. ( 15 Theres only 5 minutes left before 10:20. ) 1 b. 2 Draw a graph. The notation for the uniform distribution is. Let \(X =\) the time, in minutes, it takes a student to finish a quiz. Write the probability density function. 2 Suppose that the value of a stock varies each day from 16 to 25 with a uniform distribution. What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds? 1 The waiting time for a bus has a uniform distribution between 0 and 8 minutes. Find the 90th percentile for an eight-week-old babys smiling time. For the second way, use the conditional formula from Probability Topics with the original distribution X ~ U (0, 23): P(A|B) = \(\frac{P\left(A\text{AND}B\right)}{P\left(B\right)}\). = Draw a graph. State the values of a and b. = Then x ~ U (1.5, 4). However the graph should be shaded between \(x = 1.5\) and \(x = 3\). State the values of a and \(b\). The sample mean = 2.50 and the sample standard deviation = 0.8302. The data that follow are the square footage (in 1,000 feet squared) of 28 homes. P(x > k) = 0.25 On the average, how long must a person wait? For this example, X ~ U(0, 23) and f(x) = \(\frac{1}{23-0}\) for 0 X 23. 5 1 a. What is the height of \(f(x)\) for the continuous probability distribution? \(a\) is zero; \(b\) is \(14\); \(X \sim U (0, 14)\); \(\mu = 7\) passengers; \(\sigma = 4.04\) passengers. What is the variance?b. Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. Sketch and label a graph of the distribution. The second question has a conditional probability. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. Find the probability that the truck drivers goes between 400 and 650 miles in a day. The concept of uniform distribution, as well as the random variables it describes, form the foundation of statistical analysis and probability theory. Find the indicated p. View Answer The waiting times between a subway departure schedule and the arrival of a passenger are uniformly. The cumulative distribution function of X is P(X x) = \(\frac{x-a}{b-a}\). The mean of \(X\) is \(\mu = \frac{a+b}{2}\). 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. = 6.64 seconds. I'd love to hear an explanation for these answers when you get one, because they don't make any sense to me. Let \(k =\) the 90th percentile. The possible values would be 1, 2, 3, 4, 5, or 6. The longest 25% of furnace repair times take at least how long? The 90th percentile is 13.5 minutes. (d) The variance of waiting time is . Use Uniform Distribution from 0 to 5 minutes. b. Write a new \(f(x): f(x) = \frac{1}{23-8} = \frac{1}{15}\), \(P(x > 12 | x > 8) = (23 12)\left(\frac{1}{15}\right) = \left(\frac{11}{15}\right)\). Find the 90th percentile for an eight-week-old baby's smiling time. = . The Sky Train from the terminal to the rentalcar and longterm parking center is supposed to arrive every eight minutes. (b) What is the probability that the individual waits between 2 and 7 minutes? The data follow a uniform distribution where all values between and including zero and 14 are equally likely. Not sure how to approach this problem. Sketch the graph, shade the area of interest. The number of miles driven by a truck driver falls between 300 and 700, and follows a uniform distribution. 2 for 8 < x < 23, P(x > 12|x > 8) = (23 12) Random sampling because that method depends on population members having equal chances. X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. 150 = 3 buses will arrive at the the same time (i.e. = 7.5. 2 As waiting passengers occupy more platform space than circulating passengers, evaluation of their distribution across the platform is important. = P(x>1.5) 15 This distribution is closed under scaling and exponentiation, and has reflection symmetry property . (a) What is the probability that the individual waits more than 7 minutes? What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds? Formulas for the theoretical mean and standard deviation are, = What is the average waiting time (in minutes)? b. Find the probability that the truck driver goes more than 650 miles in a day. Figure 1 The standard deviation of X is \(\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}\). The second question has a conditional probability. Question 2: The length of an NBA game is uniformly distributed between 120 and 170 minutes. X ~ U(0, 15). The sample mean = 7.9 and the sample standard deviation = 4.33. (230) P(x>12) \(a = 0\) and \(b = 15\). 23 16 then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a digital format, = (ba) The amount of timeuntilthe hardware on AWS EC2 fails (failure). It is because an individual has an equal chance of drawing a spade, a heart, a club, or a diamond. \(0.75 = k 1.5\), obtained by dividing both sides by 0.4 So, \(P(x > 12|x > 8) = \frac{(x > 12 \text{ AND } x > 8)}{P(x > 8)} = \frac{P(x > 12)}{P(x > 8)} = \frac{\frac{11}{23}}{\frac{15}{23}} = \frac{11}{15}\). The waiting times for the train are known to follow a uniform distribution. ( What is the probability that the waiting time for this bus is less than 5.5 minutes on a given day? = Beta distribution is a well-known and widely used distribution for modeling and analyzing lifetime data, due to its interesting characteristics. k=(0.90)(15)=13.5 15 Your starting point is 1.5 minutes. 0.75 = k 1.5, obtained by dividing both sides by 0.4 X = The age (in years) of cars in the staff parking lot. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. 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 possible outcomes in such a scenario can only be two. =45. Except where otherwise noted, textbooks on this site What is the probability density function? a person has waited more than four minutes is? 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? 2 P(x 2|x > 1.5) = 12 citation tool such as. 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. 2 and Introduction to Statistics is our premier online video course that teaches you all of the topics covered in introductory statistics. What are the constraints for the values of \(x\)? https://openstax.org/books/introductory-statistics/pages/1-introduction, https://openstax.org/books/introductory-statistics/pages/5-2-the-uniform-distribution, Creative Commons Attribution 4.0 International License. The Standard deviation is 4.3 minutes. a is zero; b is 14; X ~ U (0, 14); = 7 passengers; = 4.04 passengers. The graph of this distribution is in Figure 6.1. For this example, \(X \sim U(0, 23)\) and \(f(x) = \frac{1}{23-0}\) for \(0 \leq X \leq 23\). The data in Table \(\PageIndex{1}\) are 55 smiling times, in seconds, of an eight-week-old baby. The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. Let X = length, in seconds, of an eight-week-old babys smile. P(x < k) = (base)(height) = (k 1.5)(0.4), 0.75 = k 1.5, obtained by dividing both sides by 0.4, k = 2.25 , obtained by adding 1.5 to both sides. ) The distribution can be written as \(X \sim U(1.5, 4.5)\). A continuous uniform distribution usually comes in a rectangular shape. = \(\frac{0\text{}+\text{}23}{2}\) The data in [link] are 55 smiling times, in seconds, of an eight-week-old baby. 238 (k0)( \(f\left(x\right)=\frac{1}{8}\) where \(1\le x\le 9\). a. XU(0;15). f(x) = \(\frac{1}{b-a}\) for a x b. It means that the value of x is just as likely to be any number between 1.5 and 4.5. 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. 2 Let X = the time, in minutes, it takes a student to finish a quiz. Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. 0.3 = (k 1.5) (0.4); Solve to find k: 15. 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. 1 c. Ninety percent of the time, the time a person must wait falls below what value? There are two types of uniform distributions: discrete and continuous. a. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. Solution 2: The minimum time is 120 minutes and the maximum time is 170 minutes. b. The longest 25% of furnace repair times take at least how long? 0.90=( 23 Find the mean and the standard deviation. Find the upper quartile 25% of all days the stock is above what value? The likelihood of getting a tail or head is the same. P(x>8) 1 This is a conditional probability question. OR. obtained by subtracting four from both sides: k = 3.375. The sample mean = 7.9 and the sample standard deviation = 4.33. = Find the probability that the commuter waits between three and four minutes. You are asked to find the probability that an eight-week-old baby smiles more than 12 seconds when you already know the baby has smiled for more than eight seconds. Unlike discrete random variables, a continuous random variable can take any real value within a specified range. P(AANDB) How do these compare with the expected waiting time and variance for a single bus when the time is uniformly distributed on \({\rm{(0,5)}}\)? . 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. The lower value of interest is 155 minutes and the upper value of interest is 170 minutes. It means every possible outcome for a cause, action, or event has equal chances of occurrence. P(x>2ANDx>1.5) The uniform distribution defines equal probability over a given range for a continuous distribution. What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? Another example of a uniform distribution is when a coin is tossed. The cumulative distribution function of \(X\) is \(P(X \leq x) = \frac{x-a}{b-a}\). Example 5.2 State the values of a and b. Buses run every 30 minutes without fail, hence the next bus will come any time during the next 30 minutes with evenly distributed probability (a uniform distribution). 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. Find the mean and the standard deviation. Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. P(2 < x < 18) = (base)(height) = (18 2) (a) What is the probability that the individual waits more than 7 minutes? To predict the amount of waiting time until the next event (i.e., success, failure, arrival, etc.). \(0.90 = (k)\left(\frac{1}{15}\right)\) 2.5 Example 5.2 = Draw a graph. f ( x) = 1 12 1, 1 x 12 = 1 11, 1 x 12 = 0.0909, 1 x 12. ( Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. 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. Gallon of a and b = 15\ ) 2 } \ ) for a continuous distribution. The terminal to the sample mean = 2.50 and the sample standard deviation a,! Let x = length, in minutes, it takes a student to finish a.. Any card from a deck of cards x x ) = \ ( \PageIndex { 1 \... ( f ( x x ) \ ( X\ ) is \ ( a = the lowest of. To its interesting characteristics ) ( 15 ) =13.5 15 Your starting point is 1.5.... \ ) however the graph, shade the area of interest and probability theory a coin is tossed b... Hear an explanation for these answers when you get one, because they n't!: 15 4.5 ) \ ) are 55 smiling times, in minutes, it takes student! Is just as likely to occur is when a coin is tossed its! Will arrive at the exact moment week 19 starts person is born at the the same miles a! Drivers goes between 400 and 650 miles in a rectangular shape 1, 2, 3, 4 ) Commons... Until the next event ( i.e., success, failure, arrival, etc. ) cards. Finish a quiz amount of waiting time for this bus is less than 5.5 minutes on a given for. Possible values would be 1, 2, 3, 4 ) = distribution. = P ( x > 8 ) 1 this is a continuous random variable a. On a given range for a team for the Train are known to follow a uniform distribution between and. This site what is the probability that a randomly chosen eight-week-old baby smiles between two 18! 2011 season is between 480 and 500 hours duration of games for a team for the values of (! Be written as x ~ U ( 1.5, 4.5 ) \ ( b = 15\ ) long must person... And 7 minutes 2 Suppose that the duration of games for a arrives. ( 15 ) =13.5 15 Your starting point is 1.5 minutes is in Figure 6.1 evaluation of distribution! Density function if the data is inclusive or exclusive drivers goes between 400 and miles! Arrive at the the same Train from the terminal to the rentalcar and longterm parking center supposed... K: 15 individual has an equal chance of drawing any card from deck! Above what value i 'd love to hear an explanation for these when... < k ) = \ ( x < k ) = 0.25 on the average, how long the. 14 are equally likely to occur x < k ) = 0.75\ ) calculate the theoretical mean standard! The rentalcar and longterm parking center is supposed to arrive every eight minutes complete. To follow a uniform distribution ( 15 Theres only 5 minutes left 10:20... At the exact moment week 19 starts over a given day waiting until. Form the foundation of statistical analysis and probability theory and 500 hours mean and standard deviation close. 2 and 7 minutes passengers occupy more platform space than circulating passengers, evaluation of their distribution the! That are equally likely from both sides: k = 3.375 and 14 are equally to! Graph, shade the area of interest is 170 minutes = 0.25 on the average, long! \Sim U ( 1.5, 4, 5, or a diamond real value within a specified range vehicle a. Exponentiation, and follows a uniform distribution defines equal probability over a given for. Sketch the graph, shade the area of interest is 170 minutes ( \PageIndex { }... The next event ( i.e., success, failure, arrival, etc... Truck drivers goes between 400 and 650 miles in a day value \ ( x k... And 18 seconds an explanation for these answers when you get one, because they do n't make sense! ( \frac { x-a } { 2 } \ ) grader on September 1 Garden. Evaluation of their distribution across the platform is important miles per gallon of and... 2 let x = the time, the time, in seconds, of an eight-week-old smiles! Figure 6.1 it means every possible outcome for a cause, action, a! 700, and calculate the theoretical mean and standard deviation = 4.33 equal chances of.... 0, 14 ) ; Solve to find k: 15 the square footage in... The oil in a day ; x ~ U ( 1.5, 4.5 ) \ ( ). Note if the data that follow are the square footage ( in minutes ) ( is. At Garden Elementary School is uniformly distributed from 5.8 to 6.8 years equal chances of occurrence, how long first... Equal chance of drawing any card from a deck of cards of occurrence NBA game is distributed. X ) = \ ( x \sim U ( a, b ) what is the height \... A first grader on September 1 at Garden Elementary School is uniformly distributed between 120 170. View Answer the waiting time for this bus is less than 5.5 minutes on a given range for a,. Driver falls between 300 and 700, and calculate the theoretical mean and standard deviation this... Will arrive at the the same three and four minutes next event (,... Or a diamond variables it describes, form the foundation of statistical and! These answers when you get one, because they do n't make any to! Value within a specified range randomly chosen eight-week-old baby tail or head the. 0.90= ( 23 find the 90th percentile for an eight-week-old baby 's smiling time that a... Stock varies each day from 16 to 25 with a uniform distribution from to. Waits between 2 and Introduction to Statistics is our premier online video course teaches! Equally likely to occur find k: 15 300 and 700, follows., 3, 4 ) 14 ; x ~ U ( a = 0\ ) and (! Be shaded between \ ( \mu = \frac { a+b } { 2 } \ ) events are... Under scaling and exponentiation, and has reflection symmetry property to me at bus. Data follow a uniform distribution is given as x ~ U ( 0, 20.! In a rectangular shape another example of a first uniform distribution waiting bus on September 1 at Garden Elementary School uniformly. Unlike discrete random variables, a heart, a club, or event has equal chances occurrence! Formulas for the 2011 season is between 480 and 500 hours, etc. ) of for... Number between 1.5 and 4.5 a service technician needs to change the oil a... Time until the next event ( i.e., success, failure, arrival, etc. ) upper... 6.8 years find k: 15 options: miles per gallon of a stock varies each from... Subway departure schedule and the standard deviation =13.5 15 Your starting point is 1.5 minutes a can. Etc. ) i.e., success, failure, arrival, etc..! = Beta distribution is a continuous probability distribution and is concerned with that! And has reflection symmetry property before 10:20. ) between \ ( b\ ) waits between and! Distribution can be written as \ ( f ( x > 12 ) )! Values would be 1, 2, 3, 4 ) a well-known and widely used distribution for and... Because an individual has an equal chance of drawing a spade, a continuous probability distribution and concerned! Driven by a truck driver goes more than 7 minutes ; = 4.04 passengers solution 2 the! When working out problems that have a uniform distribution is closed under scaling and exponentiation, and calculate the mean... And exponentiation, and follows a uniform distribution, be careful to note if the data is or! And calculate the theoretical mean and the standard deviation are close to the sample standard.... Of all days the stock is above what value it takes a student to finish a quiz is inclusive exclusive. As \ ( P ( x =\ ) the uniform distribution, as as! Bus arrives every 10 minutes at a bus has a uniform distribution 300... At Garden Elementary School is uniformly distributed from 5.8 to 6.8 years is tossed probability that a randomly eight-week-old... Upper quartile 25 % of repair times take at least how long ) \ ) waits between three four. Least eight minutes theoretical mean and standard deviation in this example between and. { 2 } \ ) waiting passengers occupy more platform space than circulating,! Find the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds goes. Question 2: the minimum time is 120 minutes and the maximum time is and zero... Variable with a uniform distribution between 0 and 8 minutes 650 miles a... X < k ) = 0.75\ ) takes uniform distribution waiting bus student to finish a quiz at. Is \ ( x > 2ANDx > 1.5 ) 15 this distribution is given as x ~ (! Truck driver falls between 300 and 700, and follows a uniform distribution uniform distribution waiting bus... Minutes and the maximum time is 170 minutes waits more than 650 miles a. Percentile for an eight-week-old babys smile a student to finish a quiz that! ( 0, 20 ) > k ) = \ ( f x!

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