a. 1 1.0/ 1.0 Points. ba = For the second way, use the conditional formula from Probability Topics with the original distribution X ~ U (0, 23): P(A|B) = \(\frac{P\left(A\text{AND}B\right)}{P\left(B\right)}\). 1.5+4 A distribution is given as \(X \sim U(0, 20)\). P(x>2) 4 The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. Example 5.2 2 P(B) The standard deviation of \(X\) is \(\sigma = \sqrt{\frac{(b-a)^{2}}{12}}\). Random sampling because that method depends on population members having equal chances. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. The 90th percentile is 13.5 minutes. Find the 30th percentile for the waiting times (in minutes). 2.75 The height is \(\frac{1}{\left(25-18\right)}\) = \(\frac{1}{7}\). Best Buddies Turkey Ekibi; Videolar; Bize Ulan; admirals club military not in uniform 27 ub. The McDougall Program for Maximum Weight Loss. 12 Extreme fast charging (XFC) for electric vehicles (EVs) has emerged recently because of the short charging period. 12 = 4.3. \(a\) is zero; \(b\) is \(14\); \(X \sim U (0, 14)\); \(\mu = 7\) passengers; \(\sigma = 4.04\) passengers. In reality, of course, a uniform distribution is . f(x) = \(\frac{1}{4-1.5}\) = \(\frac{2}{5}\) for 1.5 x 4. Uniform distribution can be grouped into two categories based on the types of possible outcomes. It is _____________ (discrete or continuous). At least how many miles does the truck driver travel on the furthest 10% of days? Answer: a. Can you take it from here? Develop analytical superpowers by learning how to use programming and data analytics tools such as VBA, Python, Tableau, Power BI, Power Query, and more. b. Ninety percent of the smiling times fall below the 90th percentile, \(k\), so \(P(x < k) = 0.90\), \[(k0)\left(\frac{1}{23}\right) = 0.90\]. 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. Ninety percent of the time, a person must wait at most 13.5 minutes. Find the mean, \(\mu\), and the standard deviation, \(\sigma\). P(120 < X < 130) = (130 120) / (150 100), The probability that the chosen dolphin will weigh between 120 and 130 pounds is, Mean weight: (a + b) / 2 = (150 + 100) / 2 =, Median weight: (a + b) / 2 = (150 + 100) / 2 =, P(155 < X < 170) = (170-155) / (170-120) = 15/50 =, P(17 < X < 19) = (19-17) / (25-15) = 2/10 =, How to Plot an Exponential Distribution in R. Your email address will not be published. This distribution is closed under scaling and exponentiation, and has reflection symmetry property . 3.375 hours is the 75th percentile of furnace repair times. = Births are approximately uniformly distributed between the 52 weeks of the year. Continuous Uniform Distribution - Waiting at the bus stop 1,128 views Aug 9, 2020 20 Dislike Share The A Plus Project 331 subscribers This is an example of a problem that can be solved with the. =0.7217 I thought of using uniform distribution methodologies for the 1st part of the question whereby you can do as such Question 2: The length of an NBA game is uniformly distributed between 120 and 170 minutes. =45 = Use the following information to answer the next three exercises. Pandas: Use Groupby to Calculate Mean and Not Ignore NaNs. P(x>2ANDx>1.5) What is the theoretical standard deviation? 2.5 1 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. e. \(\mu =\frac{a+b}{2}\) and \(\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}\), \(\mu =\frac{1.5+4}{2}=2.75\) Write the answer in a probability statement. f(x) = Sketch the graph, shade the area of interest. 2.5 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. Find the mean and the standard deviation. ( Creative Commons Attribution License Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. Find the third quartile of ages of cars in the lot. It would not be described as uniform probability. This is a conditional probability question. What percentile does this represent? What is the 90th percentile of square footage for homes? If you are redistributing all or part of this book in a print format, 2 Solution 3: The minimum weight is 15 grams and the maximum weight is 25 grams. The waiting time at a bus stop is uniformly distributed between 1 and 12 minute. Write the probability density function. k = 2.25 , obtained by adding 1.5 to both sides P(155 < X < 170) = (170-155) / (170-120) = 15/50 = 0.3. P(x>12ANDx>8) \(X =\) __________________. For this example, X ~ U(0, 23) and f(x) = \(\frac{1}{23-0}\) for 0 X 23. 0.125; 0.25; 0.5; 0.75; b. 238 Find the average age of the cars in the lot. = 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. Uniform Distribution. The Continuous Uniform Distribution in R. You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License. Write the answer in a probability statement. X is now asked to be the waiting time for the bus in seconds on a randomly chosen trip. 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. What is the probability that the waiting time for this bus is less than 5.5 minutes on a given day? The sample mean = 7.9 and the sample standard deviation = 4.33. Plume, 1995. P(x > 2|x > 1.5) = (base)(new height) = (4 2)\(\left(\frac{2}{5}\right)\)= ? The waiting times for the train are known to follow a uniform distribution. Draw a graph. P(x>8) Sixty percent of commuters wait more than how long for the train? Question 1: A bus shows up at a bus stop every 20 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. 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: P (x1 < X < x2) = (x2 - x1) / (b - a) where: The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. 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. \(0.3 = (k 1.5) (0.4)\); Solve to find \(k\): OR. Create an account to follow your favorite communities and start taking part in conversations. What has changed in the previous two problems that made the solutions different? 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. , it is denoted by U (x, y) where x and y are the . Financial Modeling & Valuation Analyst (FMVA), Commercial Banking & Credit Analyst (CBCA), Capital Markets & Securities Analyst (CMSA), Certified Business Intelligence & Data Analyst (BIDA), Financial Planning & Wealth Management (FPWM). 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 \(b\)). The possible outcomes in such a scenario can only be two. The longest 25% of furnace repair times take at least how long? 11 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. P(AANDB) This means that any smiling time from zero to and including 23 seconds is equally likely. Shade the area of interest. The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. 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, \[\mu = \frac{0+23}{2} = 11.50 \, seconds \nonumber\], \[\sigma = \frac{(23-0)^{2}}{12} = 6.64\, seconds. So, mean is (0+12)/2 = 6 minutes b. P(x>8) Please cite as follow: Hartmann, K., Krois, J., Waske, B. k The Uniform Distribution by OpenStaxCollege is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted. Your starting point is 1.5 minutes. \nonumber\]. What is the average waiting time (in minutes)? 12, For this problem, the theoretical mean and standard deviation are. Solution 2: The minimum time is 120 minutes and the maximum time is 170 minutes. 2 b. 41.5 Refer to [link]. 11 1 12 1 \(f(x) = \frac{1}{4-1.5} = \frac{2}{5}\) for \(1.5 \leq x \leq 4\). The second question has a conditional probability. 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. 5 Find the 90th percentile. Find the probability that he lost less than 12 pounds in the month. The probability that a randomly selected nine-year old child eats a donut in at least two minutes is _______. Formulas for the theoretical mean and standard deviation are, = = \(\frac{15\text{}+\text{}0}{2}\) X is continuous. obtained by subtracting four from both sides: k = 3.375 = The area must be 0.25, and 0.25 = (width)\(\left(\frac{1}{9}\right)\), so width = (0.25)(9) = 2.25. Then X ~ U (6, 15). Find the probability that the value of the stock is more than 19. Draw the graph of the distribution for \(P(x > 9)\). Find the 90thpercentile. d. What is standard deviation of waiting time? 5. The probability density function of X is \(f\left(x\right)=\frac{1}{b-a}\) for a x b. 15 15 The unshaded rectangle below with area 1 depicts this. 233K views 3 years ago This statistics video provides a basic introduction into continuous probability distribution with a focus on solving uniform distribution problems. 15. X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. Sketch the graph, shade the area of interest. The cumulative distribution function of X is P(X x) = \(\frac{x-a}{b-a}\). 1 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. 15+0 Find the mean and the standard deviation. What has changed in the previous two problems that made the solutions different. That is, find. 1 Find the probability that a randomly selected furnace repair requires less than three hours. Find the 90th percentile for an eight-week-old baby's smiling time. Then x ~ U (1.5, 4). 23 Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. Discrete uniform distributions have a finite number of outcomes. P(x > k) = (base)(height) = (4 k)(0.4) 23 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. \(a = 0\) and \(b = 15\). 15 \(a =\) 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 \(a < x < b\). Write a new f(x): f(x) = 2 15 = For this problem, A is (x > 12) and B is (x > 8). P(x>2) What is the theoretical standard deviation? The data in [link] are 55 smiling times, in seconds, of an eight-week-old baby. 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. 2.75 Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. However the graph should be shaded between \(x = 1.5\) and \(x = 3\). Uniform distribution refers to the type of distribution that depicts uniformity. 0.25 = (4 k)(0.4); Solve for k: \(P(x < 4) =\) _______. \(k\) is sometimes called a critical value. The sample mean = 2.50 and the sample standard deviation = 0.8302. 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. f(x) = \(\frac{1}{b-a}\) for a x b. = All values \(x\) are equally likely. What is the 90th . = 11.50 seconds and = \(\sqrt{\frac{{\left(23\text{}-\text{}0\right)}^{2}}{12}}\) \(f(x) = \frac{1}{9}\) where \(x\) is between 0.5 and 9.5, inclusive. The Standard deviation is 4.3 minutes. Find the indicated p. View Answer The waiting times between a subway departure schedule and the arrival of a passenger are uniformly. 0.10 = \(\frac{\text{width}}{\text{700}-\text{300}}\), so width = 400(0.10) = 40. What is the 90th percentile of square footage for homes? The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. Suppose it is known that the individual lost more than ten pounds in a month. )=0.8333 1 Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. ) The data in [link] are 55 smiling times, in seconds, of an eight-week-old baby. Let \(X =\) the time, in minutes, it takes a nine-year old child to eat a donut. Except where otherwise noted, textbooks on this site Note that the shaded area starts at \(x = 1.5\) rather than at \(x = 0\); since \(X \sim U(1.5, 4)\), \(x\) can not be less than 1.5. For example, in our previous example we said the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds. 3.5 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. c. This probability question is a conditional. 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. 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. a+b b. Ninety percent of the smiling times fall below the 90th percentile, k, so P(x < k) = 0.90, \(\left(\text{base}\right)\left(\text{height}\right)=0.90\), \(\text{(}k-0\text{)}\left(\frac{1}{23}\right)=0.90\), \(k=\left(23\right)\left(0.90\right)=20.7\). The waiting time for a bus has a uniform distribution between 2 and 11 minutes. Public transport systems have been affected by the global pandemic Coronavirus disease 2019 (COVID-19). Lowest value for \(\overline{x}\): _______, Highest value for \(\overline{x}\): _______. = a. 4 For this problem, \(\text{A}\) is (\(x > 12\)) and \(\text{B}\) is (\(x > 8\)). For this example, \(X \sim U(0, 23)\) and \(f(x) = \frac{1}{23-0}\) for \(0 \leq X \leq 23\). Second way: Draw the original graph for X ~ U (0.5, 4). Let X = the time, in minutes, it takes a student to finish a quiz. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. P(x > k) = 0.25 Find the mean, , and the standard deviation, . b. = \(\frac{a\text{}+\text{}b}{2}\) The graph of the rectangle showing the entire distribution would remain the same. Use the following information to answer the next ten questions. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . 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. 16 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. Find the probability that a random eight-week-old baby smiles more than 12 seconds KNOWING that the baby smiles MORE THAN EIGHT SECONDS. are licensed under a, 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, Mathematical Phrases, Symbols, and Formulas, Notes for the TI-83, 83+, 84, 84+ Calculators. The 30th percentile of repair times is 2.25 hours. This means that any smiling time from zero to and including 23 seconds is equally likely. With continuous uniform distribution, just like discrete uniform distribution, every variable has an equal chance of happening. We are interested in the weight loss of a randomly selected individual following the program for one month. The graph illustrates the new sample space. 5 Correct me if I am wrong here, but shouldn't it just be P(A) + P(B)? The uniform distribution defines equal probability over a given range for a continuous distribution. 23 The percentage of the probability is 1 divided by the total number of outcomes (number of passersby). obtained by subtracting four from both sides: k = 3.375. 14.6 - Uniform Distributions. = (Hint the if it comes in the first 10 minutes and the last 15 minutes, it must come within the 5 minutes of overlap from 10:05-10:10. = It explains how to. P(x < k) = (base)(height) = (k 1.5)(0.4) Sketch the graph, and shade the area of interest. a. A bus arrives every 10 minutes at a bus stop. 2.1.Multimodal generalized bathtub. The interval of values for \(x\) is ______. It is because an individual has an equal chance of drawing a spade, a heart, a club, or a diamond. c. Ninety percent of the time, the time a person must wait falls below what value? percentile of this distribution? P(AANDB) Let X = the time, in minutes, it takes a student to finish a quiz. b. Ninety percent of the smiling times fall below the 90th percentile, k, so P(x < k) = 0.90. . Another example of a uniform distribution is when a coin is tossed. 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. 11 k=(0.90)(15)=13.5 A continuous probability distribution is a Uniform distribution and is related to the events which are equally likely to occur. The data that follow are the number of passengers on 35 different charter fishing boats. They can be said to follow a uniform distribution from one to 53 (spread of 52 weeks). 2 )( ) hours and For this reason, it is important as a reference distribution. b. Find the 90th percentile. Sketch the graph of the probability distribution. ) Learn more about us. \(P(x > k) = 0.25\) 0.625 = 4 k, 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. = Then \(x \sim U(1.5, 4)\). Find the probability that the time is at most 30 minutes. Your starting point is 1.5 minutes. a. The shaded rectangle depicts the probability that a randomly. So, P(x > 12|x > 8) = A student takes the campus shuttle bus to reach the classroom building. 2 Let \(X =\) length, in seconds, of an eight-week-old baby's smile. Solution: 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). for 0 X 23. 5 . The waiting time at a bus stop is uniformly distributed between 1 and 12 minute. Solve the problem two different ways (see Example). 1 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. hours and \(\sigma =\sqrt{\frac{{\left(41.5\right)}^{2}}{12}}=0.7217\) hours. Another simple example is the probability distribution of a coin being flipped. What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours? 1.5+4 Find the probability that the time is more than 40 minutes given (or knowing that) it is at least 30 minutes. \(P\left(x
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