Then X ~ U (6, 15). Find the mean, Ninety percent of the time, the time a person must wait falls below what value? The notation for the uniform distribution is X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. Find the 90th percentile. All values x are equally likely. Let X = the time needed to change the oil on a car. Let X = the time, in minutes, it takes a nine-year old child to eat a donut. Find the probability that a randomly selected furnace repair requires less than three hours. You must reduce the sample space. It is used to. This means that any smiling time from zero to and including 23 seconds is equally likely. a = smallest X; b = largest X, The mean is [latex]\displaystyle\mu=\frac{{{a}+{b}}}{{2}}\\[/latex], The standard deviation is [latex]\displaystyle\sigma=\sqrt{{\frac{{({b}-{a})}^{{2}}}{{12}}}}\\[/latex], Probability density function: [latex]\displaystyle{f{{({x})}}}=\frac{{1}}{{{b}-{a}}} \text{ for } {a}\leq{X}\leq{b}\\[/latex], Area to the Left of x: [latex]\displaystyle{P}{({X}{<}{x})}={({x}-{a})}{(\frac{{1}}{{{b}-{a}}})}\\[/latex], Area to the Right of x: [latex]\displaystyle{P}{({X}{>}{x})}={({b}-{x})}{(\frac{{1}}{{{b}-{a}}})}\\[/latex], Area Between c and d: [latex]\displaystyle{P}{({c}{<}{x}{<}{d})}={(\text{base})}{(\text{height})}={({d}-{c})}{(\frac{{1}}{{{b}-{a}}})}\\[/latex], [latex]\displaystyle{P}{({x}{<}{k})}={(\text{base})}{(\text{height})}={({12.5}-{0})}{(\frac{{1}}{{15}})}={0.8333}\\[/latex], [latex]\displaystyle{P}{({x}{>}{2}|{x}{>}{1.5})}={(\text{base})}{(\text{new height})}={({4}-{2})}{(\frac{{2}}{{5}})}=\frac{{4}}{{5}}\\[/latex], http://cnx.org/contents/30189442-6998-4686-ac05-ed152b91b9de@17.41:36/Introductory_Statistics, http://cnx.org/contents/30189442-6998-4686-ac05-ed152b91b9de@17.44. For example, it can arise in inventory management Auditing InventoryAuditing inventory is the process of cross-checking financial records with physical inventory and records. The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution. It is important to practice examples of uniform distribution after learning it’s formulas. Discrete uniform distribution is also useful in Monte Carlo simulationMonte Carlo SimulationMonte Carlo simulation is a statistical method applied in modeling the probability of different outcomes in a problem that cannot be simply solved, due to the interference of a random variable.. Another simple example is the probability distribution of a coin being flipped. Find the 30th percentile of furnace repair times. Moreover, statistics concepts can help investors monitor, The normal distribution is also referred to as Gaussian or Gauss distribution. The domain is a finite interval. Monte Carlo simulation is often used to forecast scenarios and help in the identification of risks. However, there is an infinite number of points that can exist. P(2 < x < 18) = 0.8; 90th percentile = 18. Find the mean and the standard deviation. The mean of X is [latex]\displaystyle{\mu}=\frac{{{a}+{b}}}{{2}}\\[/latex]. Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. Uniform Distribution has a constant probability. The possible values would be 1, 2, 3, 4, 5, or 6. The probability density function is [latex]\displaystyle{f{{({x})}}}=\frac{{1}}{{{b}-{a}}}\\[/latex] for a ≤ x ≤ b. Draw a graph. A good example of a continuous uniform distribution is an idealized random number generator. Examples of Uniform Distribution. Therefore, each time the 6-sided die is thrown, each side has a chance of 1/6. In this case, each of the six numbers has an equal chance of appearing. X = a real number between a and b (in some instances, X can take on the values a and b). a is zero; b is 14; X ~ U (0, 14); μ = 7 passengers; σ = 4.04 passengers. A continuous uniform distribution (also referred to as rectangular distribution) is a statistical distribution with an infinite number of equally likely measurable values. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. The total duration of baseball games in the major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive. The data follow a uniform distribution where all values between and including zero and 14 are equally likely. It is also known as rectangular distribution. Let X = length, in seconds, of an eight-week-old baby’s smile. A form of probability distribution where every possible outcome has an equal likelihood of happening, Auditing inventory is the process of cross-checking financial records with physical inventory and records. The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer). In this Example we use Chebfun to solve two problems involving the uniform distribution from the textbook [1]. OpenStax, Statistics, The Uniform Distribution. CFI is the official provider of the global Financial Modeling & Valuation Analyst (FMVA)™FMVA® CertificationJoin 350,600+ students who work for companies like Amazon, J.P. Morgan, and Ferrari certification program, designed to help anyone become a world-class financial analyst. The sample mean = 7.9 and the sample standard deviation = 4.33. For example, if you stand on a street corner and start to randomly hand a $100 bill to any lucky person who walked by, then every passerby would have an equal chance of being handed the money. If X has a uniform distribution where a < x < b or a ≤ x ≤ b, then X takes on values between a and b (may include a and b). Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. The probability is constant since each variable has equal chances of being the outcome. We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. Another example with a uniform distribution is when a coin is tossed. 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. The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. 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. 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. The possible outcomes in such a scenario can only be two. The graph of a uniform distribution is usually flat, whereby the sides and top are parallel to the x- and y-axes. For this example, X ~ U(0, 23) and [latex]\displaystyle{f{{({x})}}}=\frac{{1}}{{{23}-{0}}}\\[/latex] for 0 ≤ X ≤ 23. The McDougall Program for Maximum Weight Loss. Uniform distribution can be grouped into two categories based on the types of possible outcomes. Let X= leng… Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. We write X ∼ U(a, b). The data in the table below are 55 smiling times, in seconds, of an eight-week-old baby. There are several ways in which discrete uniform distribution can be valuable for businesses. Formulas for the theoretical mean and standard deviation are [latex]\displaystyle{\mu}=\frac{{{a}+{b}}}{{2}}{\quad\text{and}\quad}{\sigma}=\sqrt{{\frac{{{({b}-{a})}^{{2}}}}{{12}}}}\\[/latex], For this problem, the theoretical mean and standard deviation are [latex]\displaystyle{\mu}=\frac{{{0}+{23}}}{{2}}={11.50} \text{ seconds}{\quad\text{and}\quad}{\sigma}=\sqrt{{\frac{{{({23}-{0})}^{{2}}}}{{12}}}}={6.64} \text{ seconds}\\[/latex]. State the values of a and b. Let. The sample mean = 11.49 and the sample standard deviation = 6.23. Monte Carlo simulation is a statistical method applied in modeling the probability of different outcomes in a problem that cannot be simply solved, due to the interference of a random variable. Not all uniform distributions are discrete; some are continuous. On the average, how long must a person wait? Solve the problem two different ways (see Example 3). A deck of cards also has a uniform distribution. It is impossible to get a value of 1.3, 4.2, or 5.7 when rolling a fair die. This question has a conditional probability. McDougall, John A. [latex]\displaystyle{\mu}=\frac{{{a}+{b}}}{{2}}=\frac{{{15}+{0}}}{{2}}={7.5}\\[/latex]. Plume, 1995. A distribution is given as X ~ U (0, 20). 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. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. [latex]\displaystyle{\sigma}=\sqrt{{\frac{{{({b}-{a})}^{{2}}}}{{12}}}}=\sqrt{{\frac{{{({15}-{0})}^{{2}}}}{{12}}}}={4.3}\\[/latex]The standard deviation is 4.3 minutes. The data that follow are the number of passengers on 35 different charter fishing boats.