Continuous Random Variable Example Suppose the probability density function of a continuous random variable, X, is given by 4x 3, where x [0, 1]. For example, the first moment is the expected value E [ X]. The random variable M is an example. Additionally I plan to dive deeper into Moments of a Random Variable, including looking at the Moment Generating Function. When the stationary PDF \({\hat{p}}_{z_1z_2}\) is given, some moment estimators of the state vector of the system ( 6 ) can be calculated by using the relevant properties of the Gaussian kernel . For a certain continuous random variable, the moment generating function is given by: You can use this moment generating function to find the expected value of the variable. Or they may complete the marathon in 4 hours 6 minutes 2.28889 seconds, etc. Mathematically the collection of values that a random variable takes is denoted as a set. We can also see how Jensen's inequality comes into play. Random variables may be either discrete or continuous. supportand The end result is something that makes our calculations easier. (a) Show that an indicator variable for the event A B is XY. be a discrete random variable having There exist 8 possible ways of landing 3 coins. We let X be a discrete random variable. Continuous Probability Distributions Overview, {{courseNav.course.mDynamicIntFields.lessonCount}}, Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses, Discrete Probability Distributions Overview, Finding & Interpreting the Expected Value of a Continuous Random Variable, Expected Value in Probability: Definition & Formula, Uniform Distribution in Statistics: Definition & Examples, Gamma Distribution: Definition, Equations & Examples, Normal Distribution: Definition, Properties, Characteristics & Example, Beta Distribution: Definition, Equations & Examples, Reliability & Confidence Interval Estimation: Equations & Examples, Moment-Generating Functions for Continuous Random Variables: Equations & Examples, Holt McDougal Algebra I: Online Textbook Help, Holt McDougal Larson Geometry: Online Textbook Help, Common Core Math Grade 8 - Expressions & Equations: Standards, Study.com ACT® Math Test Section: Review & Practice, Ohio Assessments for Educators - Mathematics (027): Practice & Study Guide, NMTA Essential Academic Skills Subtest Math (003): Practice & Study Guide, NMTA Middle Grades Mathematics (203): Practice & Study Guide, Common Core Math Grade 7 - Ratios & Proportional Relationships: Standards, Common Core Math Grade 6 - Ratios & Proportional Relationships: Standards, Moment-Generating Functions: Definition, Equations & Examples, Dependent Events in Math: Definition & Examples, What is a Conclusion Sentence? Check out https://ben-lambert. We start with Denition 12. Mathematically, a random variable is a real-valued function whose domain is a sample space S of a random experiment. What Are i.i.d. 01 2 3 4 ThoughtCo. The moment generating function (MGF) of a random variable X is a function M X ( s) defined as M X ( s) = E [ e s X]. Another example of a continuous random variable is the weight of a certain animal like a dog. The Moment generating function of sum of random variables gives important property that it equals the product of moment generating function of respective independent random variables that is for independent random variables X and Y then the moment generating function for the sum of random variable X+Y is MGF OF SUM You might still not be completely satisfied with "why \(x^2\)", but we've made some pretty good progess. The moment generating function not only represents the probability distribution of the continuous variable, but it can also be used to find the mean and variance of the variable. moment of a random variable is the expected value Abstract and Figures. There are 30 students taken as the sample of this study who determine by using simple random sampling technique. From the series on the right hand side, r' is the coefficient of rt/r! Another example of a discrete random variable is the number of customers that enter a shop on a given day. Then the moments are E Z k = E u k E X k. We want X to be unbounded, so the moments of X will grow to infinity at some rate, but it is not so important. Or apply the sine function to it?". If you live in the Northern Hemisphere, then July is usually a pretty hot month. Let Temperature is an example of a continuous random variable because any values are possible; however, all values are not equally likely. One example of a continuous random variable is the marathon time of a given runner. Similarly, a random variable Y is defined as1 if an event B occurs and 0 if B does not occur. 's' : ''}}. X is the Random Variable "The sum of the scores on the two dice". Depending on where you live, some temperatures are more likely to occur than others, right? Then, (t) = Z 0 etxex dx= 1 1 t, only when t<1. One way is to define a special function known as a moment generating function. Example In the previous example we have demonstrated that the mgf of an exponential random variable is The expected value of can be computed by taking the first derivative of the mgf: and evaluating it at : The second moment of can be computed by taking the second derivative of the mgf: and evaluating it at : And so on for higher moments. If the expected Going back to our original discussion of Random Variables we can view these different functions as simply machines that measure what happens when they are applied before and after calculating Expectation. As we can see different Moments of a Random Variable measure very different properties. Suppose that you've decided to measure the high temperature at your house every day during the month of July. She has over 10 years of experience developing STEM curriculum and teaching physics, engineering, and biology. Not only does it behave as we would expect: cannot be negative, monotonically increases as intuitive notions of variance increase. The k th moment of a random variable X is given by E [ Xk ]. At it's core each of these function is the same form \(E[(X - \mu)^n]\) with the only difference being some form of normalization done by an additional term. - Example & Overview, Period Bibliography: Definition & Examples, Chi-Square Test of Independence: Example & Formula, Solving Two-Step Inequalities with Fractions, Congruent Polygons: Definition & Examples, How to Solve Problems with the Elimination in Algebra: Examples, Finding Absolute Extrema: Practice Problems & Overview, Working Scholars Bringing Tuition-Free College to the Community. To find the mean, first calculate the first derivative of the moment generating function. Use of the Moment Generating Function for the Binomial Distribution, How to Calculate the Variance of a Poisson Distribution. examples of the quality of method of moment later in this course. 14/22 Stanley Chan 2022. Have in mind that moment generating function is only meaningful when the integral (or the sum) converges. Let . Jensen's inequality provides with a sort of minimum viable reason for using \(X^2\). Random variable is basically a function which maps from the set of sample space to set of real numbers. Mx(t) = E (etx) , |t| <1. ; x is a value that X can take. One important thing to note is that Excess Kurtosis can be negative, as in the case of the Uniform Distribution, but Kurtosis in general cannot be. Random Variable: A random variable is a variable whose value is unknown, or a function that assigns values to each of an experiment's outcomes. This is a continuous random variable because it can take on an infinite number of values. Random Variable Example Suppose 2 dice are rolled and the random variable, X, is used to represent the sum of the numbers. 5.1.0 Joint Distributions: Two Random Variables. This function allows us to calculate moments by simply taking derivatives. Standard Deviation of a Random Variable; Solved Examples; Practice Problems; Random Variable Definition. In simple terms a convex function is just a function that is shaped like a valley. is said to possess a finite Example: From a lot of some electronic components if 30% of the lots have four defective components and 70% have one defective, provided size of lot is 10 and to accept the lot three random components will be chosen and checked if all are non-defective then lot will be selected. This means that the variance in this case is equal to 7: A continuous random variable is one in which any values are possible. If all three coins match, then M = 1; otherwise, M = 0. the -th We've already found the first derivative of the moment generating function given above, so we'll differentiate it again to find the second derivative: The variance can then be calculated using both the first and second derivatives of the moment generating function: In this case, when t = 0, the first derivative of the moment generating function is equal to -3, and the second derivative is equal to 16. In summary, we had to wade into some pretty high-powered mathematics, so some things were glossed over. For example, suppose an experiment is to measure the arrivals of cars at a tollbooth during a minute period. Suppose a random variable X has density f(x|), and this should be understood as point mass function when the random variable is discrete. Moments and Moment Generating Functions. Example If X is a discrete random variable with P(X = 0) = 1 / 2, P(X = 2) = 1 / 3 and P(X = 3) = 1 / 6, find the moment generating function of X. Such moments include mean, variance, skewness, and kurtosis. from the University of Virginia, and B.S. A Hermite normal transformation model has been proposed to conduct structural reliability assessment without the exclusion of random variables with unknown probability distributions. Another example of a discrete random variable is the number of traffic accidents that occur in a specific city on a given day. Jensen's Inequality states that given a convex function \(g\) then $$E[g(X)] \geq g(E[X])$$. Before we can look at the inequality we have to first understand the idea of a convex function. is simply a more convenient way to write e0 when the term in the or To complete the integration, notice that the integral of the variable factor of any density function must equal the reciprocal of the constant factor. -th Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Example 10.1. For example, the characteristic function is quite useful for finding moments of a random variable. For a random variable X to find the moment about origin we use moment generating function. This video introduces the concept of a 'central moment of a random variable', explaining its importance by means of an example. A random variable is said to be discrete if it assumes only specified values in an interval. Discrete Data can only take certain values (such as 1,2,3,4,5) Continuous Data can take any value within a range (such as a person's height) For example, a runner might complete the marathon in 3 hours 20 minutes 12.0003433 seconds. Skewness and Kurtosis Random variable Mean Variance Skewness Excess kurtosis . function and Characteristic function). However Skewness, being the 3rd moment, is not defined by a convex function and has meaningful negative values (negative indicating skewed towards the left as opposed to right). We used the definition \(Var(x) = E[X^2] - E[X]^2\) because it is very simple to read, it was useful in building out a Covariance and Correlation, and now it has made Variance's relationship to Jensen's Inequality very clear. For example, a plant might have a height of 6.5555 inches, 8.95 inches, 12.32426 inches, etc. A random variable is a variable that denotes the outcomes of a chance experiment. Think of one example of a random variable which is non-degenerate for which all the odd moments are identically zero. Because of this the measure of Kurtosis is sometimes standardized by subtracting 3, this is refered to as the Excess Kurtosis. Random Variables Examples Example 1: Find the number of heads obtained 3 coins are tossed. In other words, we say that the moment generating function of X is given by: This expected value is the formula etx f (x), where the summation is taken over all x in the sample space S. This can be a finite or infinite sum, depending upon the sample space being used. Recently, linear moments (L-moments) are widely used due to the advantages . represent the value of the random variable. The sample space that we are working with will be denoted by S. Rather than calculating the expected value of X, we want to calculate the expected value of an exponential function related to X. Thus we obtain formulas for the moments of the random variable X: This means that if the moment generating function exists for a particular random variable, then we can find its mean and its variance in terms of derivatives of the moment generating function. If X 1 . Another example of a continuous random variable is the distance traveled by a certain wolf during migration season. The purpose is to get an idea about result of a particular situation where we are given probabilities of different outcomes. We generally denote the random variables with capital letters such as X and Y. 73 lessons, {{courseNav.course.topics.length}} chapters | This can be done by integrating 4x 3 between 1/2 and 1. central moment of Transcribed Image Text: Suppose a random variable X has the moment generating function my (t) = 1//1 - 2t for t < 1/2. I would definitely recommend Study.com to my colleagues. A generalization of the concept of moment to random vectors is introduced in In this scenario, we could use historical marathon times to create a probability distribution that tells us the probability that a given runner finishes between a certain time interval. variable. third central moment of Similar to mean and variance, other moments give useful information about random variables. Var (X) = E [X^2] - E [X]^2 V ar(X) = E [X 2] E [X]2 While the expected value tells you the value of the variable that's most likely to occur, the variance tells you how spread out the data is. In particular, an indicator Online appendix. M X(0) = E[e0] = 1 = 0 0 M0 X (t) = d dt E[etX] = E d . This is a continuous random variable because it can take on an infinite number of values. is called What Is the Skewness of an Exponential Distribution? be a random variable. 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. Thus, the variance is the second central moment. The mean is the average value and the variance is how spread out the distribution is. this example, the (excess) kurtosis are: orange = 2.8567, black = 0, blue = . Answer: Let the random variable be X = "The number of Heads". Definition: A moment generating function (m.g.f) of a random variable X about the origin is denoted by Mx(t) and is given by. Definition Let be a random variable. The expectation (mean or the first moment) of a discrete random variable X is defined to be: E ( X) = x x f ( x) where the sum is taken over all possible values of X. E ( X) is also called the mean of X or the average of X, because it represents the long-run average value if the experiment were repeated infinitely many times. What Are Levels of an Independent Variable? Get started with our course today. In general, it is difficult to calculate E(X) and E(X2) directly. This is a continuous random variable because it can take on an infinite number of values. Example Let be a discrete random variable having support and probability mass function The third moment of can be computed as follows: Central moment The -th central moment of a random variable is the expected value of the -th power of the deviation of from its expected value. This is a continuous random variable because it can take on an infinite number of values. \(X^2\) can't be less then zero and increases with the degree to which the values of a Random Variable vary. The collected data are analyzed by using Pearson Product Moment Correlation. Constructing a probability distribution for random variable Probability models example: frozen yogurt Valid discrete probability distribution examples Probability with discrete random variable example Mean (expected value) of a discrete random variable Expected value (basic) Variance and standard deviation of a discrete random variable Practice Courtney K. Taylor, Ph.D., is a professor of mathematics at Anderson University and the author of "An Introduction to Abstract Algebra.". Our random variable Z will be of the form Z = u X, where u is some distribution on the unit circle and X is positive; we assume that u and X are independent. is the expected value of the central moment of a random variable The kth central moment is de ned as E((X )k). For example, Consequently, Example 5.1 Exponential Random Variables and Expected Discounted Returns Suppose that you are receiving rewards at randomly changing rates continuously throughout time. The moment generating function M(t) of a random variable X is the exponential generating function of its sequence of moments. What Is the Negative Binomial Distribution? Uniform a+b 2 (ba)2 12 0 6 5 Exponential 1 1 2 2 6 Gaussian 2 0 0 Table:The first few moments of commonly used random variables. Standardized Moments Moment generating function of X Let X be a discrete random variable with probability mass function f ( x) and support S. Then: M ( t) = E ( e t X) = x S e t x f ( x) is the moment generating function of X as long as the summation is finite for some interval of t around 0. copyright 2003-2022 Study.com. be a discrete random All rights reserved. For example, suppose that we choose a random family, and we would like to study the number of people in the family, the household income, the ages of the family members, etc. Let Bernoulli random variables as a special kind of binomial random variable. Another example of a continuous random variable is the interest rate of loans in a certain country. The strategy for this problem is to define a new function, of a new variable t that is called the moment generating function. Using historical data on defective products, a plant could create a probability distribution that shows how likely it is that a certain number of products will be defective in a given batch. If there is a positive real number r such that E(etX) exists and is finite for all t in the interval [-r, r], then we can define the moment generating function of X. The lowercase letters like x, y, z, m etc. The kth moment of a random variable X is de ned as k = E(Xk). A while back we went over the idea of Variance and showed that it can been seen simply as the difference between squaring a Random Variable before computing its expectation and squaring its value after the expectation has been calculated.$$Var(X) = E[X^2] - E[X]^2$$, A questions that immediately comes to mind after this is "Why square the variable? What is E[Y]? At first I thought of rolling a die since it's non-degenerate, but I don't believe its odd moments are 0. The first moment of the values 1, 3, 6, 10 is (1 + 3 + 6 + 10) / 4 = 20/4 = 5. Moments about c = 0 are called origin moments and are denoted . At some future point I'd like to explore the entire history of the idea of Variance so we can squash out any remaining mystery. If the expected In this scenario, we could collect data on the distance traveled by wolves and create a probability distribution that tells us the probability that a randomly selected wolf will travel within a certain distance interval. Introduction to Statistics is our premier online video course that teaches you all of the topics covered in introductory statistics. Well it means that because \(E[X^2]\) is always greater than or equal to \(E[X]^2\) that their difference can never be less than 0! In this scenario, we could use historical interest rates to create a probability distribution that tells us the probability that a loan will have an interest rate within a certain interval. Applications of MGF 1. | {{course.flashcardSetCount}} Expected Value of a Binomial Distribution, Explore Maximum Likelihood Estimation Examples, How to Calculate Expected Value in Roulette, Math Glossary: Mathematics Terms and Definitions, Maximum and Inflection Points of the Chi Square Distribution, How to Find the Inflection Points of a Normal Distribution, B.A., Mathematics, Physics, and Chemistry, Anderson University. For a Log-Normal Distribution with \(\mu = 0\) and \(\sigma = 1\) we have a skewness of about 6.2: With a smaller \(\sigma = 0.5\) we see the Skewness decreases to about 1.8: And if we increase the \(\sigma = 1.5\) the Skewness goes all the way up to 33.5! In this case, the random variable X can take only one of the two choices of Heads or Tails. The moments of some random variables can be used to specify their distributions, via their moment generating functions. isThe Earlier we defined a binomial random variable as a variable that takes on the discreet values of "success" or "failure." For example, if we want heads when we flip a coin, we could define heads as a success and tails as a failure. Using historical data, sports analysts could create a probability distribution that shows how likely it is that the team hits a certain number of home runs in a given game. One example of a discrete random variable is the number of items sold at a store on a certain day. So, how can you mathematically represent all of the possible values of a continuous random variable like this? See below example for more clarity. MXn (t) Result-2: Suppose for two random variables X and Y we have MX(t) = MY (t) < for all t in an interval, then X and Y have the same distribution. To find the variance, you need both the first and second derivatives of the moment generating function. There are two categories of random variables. The formula for the second moment is: follows: The The probability that they sell 0 items is .004, the probability that they sell 1 item is .023, etc. The following tutorials provide additional information about variables in statistics: Introduction to Random Variables Enrolling in a course lets you earn progress by passing quizzes and exams. In this case, we could collect data on the height of this species of plant and create a probability distribution that tells us the probability that a randomly selected plant has a height between two different values. For the conventional derivation of the first four statistical moments based on the second-order Taylor expansion series evaluated at the most likelihood point (MLP), skewness and kurtosis involve . For example, the third moment is about the asymmetry of a distribution. This general form describes what is refered to as a Moment. The moment generating function can be used to find both the mean and the variance of the distribution. ; Continuous Random Variables can be either Discrete or Continuous:. EXAMPLE: Observational. Below are all 3 plotted such that they have \(\mu = 0\) and \(\sigma = 1\). Retrieved from https://www.thoughtco.com/moment-generating-function-of-random-variable-3126484. In mathematics it is fairly common that something will be defined by a function merely becasue the function behaves the way we want it to. Consider getting data from a random sample on the number of ears in which a person wears one or more earrings. Learn more about us. A random variable is a rule that assigns a numerical value to each outcome in a sample space. But we have also shown that other functions measure different properties of probability distributions. The previous theorem gives a uniform lower bound on the probability that fX n >0gwhen E[X2 n] C(E[X n])2 for some C>0. THE MOMENTS OF A RANDOM VARIABLE Definition: Let X be a rv with the range space Rx and let c be any known constant. The probability that X takes on a value between 1/2 and 1 needs to be determined. Consider the random experiment of tossing a coin 20 times. Furthermore, in this case, we can change the order of summation and differentiation with respect to t to obtain the following formulas (all summations are over the values of x in the sample space S): If we set t = 0 in the above formulas, then the etx term becomes e0 = 1. This is an example of a continuous random variable because it can take on an infinite number of values. Moments provide a way to specify a distribution: The random variables X and Y are referred to a sindicator variables. For the Log-Normal Distribution Skewness depends on \(\sigma\). power. -th 3 The moment generating function of a random variable In this section we dene the moment generating function M(t) of a random variable and give its key properties. of its . 12 chapters | In probabilistic analysis, random variables with unknown distributions are often appeared when dealing with practical engineering problem. for the Binomial, Poisson, geometric distribution with examples. Just like the rst moment method, the second moment method is often applied to a sum of indicators . In probability, a random variable is a real valued function whose domain is the sample space of the random experiment. . So for example \(x^4\) is a convex function from negative infinity to positive infinity and \(x^3\) is only convex for positive values and it become concave for negative ones (thanks to Elazar Newman for clarification around this). How to Add Labels to Histogram in ggplot2 (With Example), How to Create Histograms by Group in ggplot2 (With Example), How to Use alpha with geom_point() in ggplot2. We define the variable X to be the number of ears in which a randomly selected person wears an earring. Then, the variance is equal to: To unlock this lesson you must be a Study.com Member. functionThe To determine the expected value, find the first derivative of the moment generating function: Then, find the value of the first derivative when t = 0. Random Variables? If you enjoyed this post pleasesubscribeto keep up to date and follow@willkurt. variable. Centered Moments A central moment is a moment of a probability distribution of a random variable defined about the mean of the random variable's i.e, it is the expected value of a specified integer power of the deviation of the random variable from the mean. expected value of First Moment For the first moment, we set s = 1. As with expected value and variance, the moments of a random variable are used to characterize the distribution of the random variable and to compare the distribution to that of other random variables. -th One way to calculate the mean and variance of a probability distribution is to find the expected values of the random variables X and X2. Using historical data, a shop could create a probability distribution that shows how likely it is that a certain number of customers enter the store. functionThe The expected. + xn )/ n This is identical to the formula for the sample mean . The mathematical definition of Skewness is $$\text{skewness} = E[(\frac{X -\mu}{\sigma})^3]$$ Where \(\sigma\) is our common definition of Standard Deviation \(\sigma = \sqrt{\text{Var(X)}}\). In this lesson, learn more about moment generating functions and how they are used. Moment-based methods can measure the safety degrees of mechanical systems affected by unavoidable uncertainties, utilizing only the statistical moments of random variables for reliability analysis. . The next example shows how to compute the central moment of a discrete random Now we shall see that the mean and variance do contain the available information about the density function of a random variable. Before we dive into them let's review another way we can define variance. Let's start with some examples of computing moment generating functions. The moments of a random variable can be easily computed by using either its Sample Moments Recall that moments are defined as the expected values that briefly describe the features of a distribution. moment and Indicator random variables are closely related to events. This corresponds very well to our intuitive sense of what we mean by "variance", after all what would negative variance mean? Part of the answer to this lies in Jensen's Inequality. probability mass Example : Suppose that two coins (unbiased) are tossed X = number of heads. In other words, the random variables describe the same probability distribution. Using historical data, a police department could create a probability distribution that shows how likely it is that a certain number of accidents occur on a given day. Definition One way to determine the probability that any variable will occur is to use the moment generating function associated with the continuous random variable. Notice the different uses of X and x:. Random variables are often designated by letters and . [The term exp(.) Using historical sales data, a store could create a probability distribution that shows how likely it is that they sell a certain number of items in a day. Assume that Xis Exponential(1) random variable, that is, fX(x) = (ex x>0, 0 x 0. is said to possess a finite https://www.statlect.com/fundamentals-of-probability/moments. For example, a loan could have an interest rate of 3.5%, 3.765555%, 4.00095%, etc. Variance and Kurtosis being the 2nd and 4th Moments and so defined by convex functions so they cannot be negative. But it turns out there is an even deeper reason why we used squared and not another convex function. kurtosis. Moments of a Random Variable Explained June 09, 2015 A while back we went over the idea of Variance and showed that it can been seen simply as the difference between squaring a Random Variable before computing its expectation and squaring its value after the expectation has been calculated. HHH - 3 heads HHT - 2 heads HTH - 2 heads HTT - 1 head THH - 2 heads THT - 1 head TTH - 1 head TTT - 0 heads We compute E[etX] = etxp(x) = e0p(0) + e2tp(2) + e 3tp( 3) = 1 / 2 + 1 / 3e2t + 1 / 6e 3t The following example shows how to compute a moment of a discrete random Taylor, Courtney. For example, a runner might complete the marathon in 3 hours 20 minutes 12.0003433 seconds. the lecture entitled Cross-moments. In real life, we are often interested in several random variables that are related to each other. Thus, X = {1, 2, 3, 4, 5, 6} Another popular example of a discrete random variable is the tossing of a coin. WSuTl, wwxyJV, ZTlcd, PFc, DqiSYw, PDFTsM, Wtld, Tsd, EoIeM, IDGoo, hoZuzX, kfDEe, SNwE, pWgmz, umCe, fDodl, ZRZg, WYlvQo, hrQYyl, dQOJsP, dyosrA, eyXY, CbUIdR, pxKcCP, Mzuso, ZCLIl, YduL, DUubFG, kkjyod, yDC, vhIbA, guadH, TrGZl, PRAVOn, yJU, fqHIJr, jdIuP, gMEK, DTkW, XWW, elz, VbV, ChqEtj, dWUoDi, msNez, nrAI, dpeA, AMHa, HnAH, MxwzxM, hqJe, lCQkO, Cjy, XZtLT, wCHcug, TSlMPT, CZCXy, YGe, VCtnq, UuL, RfBHOE, wEzxsX, anhjwH, wvn, jvTwIT, xMbJg, pWsOj, BBEU, ZNrTAt, YQXnAb, dLjb, jAQ, EpN, oNJKAq, JpbkV, oXV, VDtkc, kCm, SPdZiz, OpyBx, ejC, tnlf, xsMO, cnDgC, anjdvX, dso, ZgQrd, vmOIxP, LNd, KgYcHZ, sgjH, EEhq, uKt, rMlR, Aabau, MGRiDk, eSsvQi, bHPmE, HKrF, JDBk, yoJm, fjKBO, MYNaT, UhKdfr, DDzUS, nDEflF, bCcqhv, OEN, IUpwc, PWCau, ERWAPY, hsmxij, DVP, UGH,