Consider the truncated form of the 3-parameter and 3-parameter Beta distribution between 0.2 and 0.8 as a distribution (as in the composition and accept-reject techniques, you need to divide by its area). The probability density function is given as: 2,4 2,2 2,0 1,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0,0 0 I f(x) = 30x² (1-x)².0 x≤1 0,2 0,4 0,6 a) Using the Kolmogorov-Smirnov test, examine whether the following data come from the given truncated probability distribution. 0.36-0.57-0.79-0.21 -0.67 -0.46 ) Develop an accept-reject method using the given trapezoidal dominant function to generate random values from this distribution. Obtain two random values using this method.
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- The Gamma distribution is often used to describe the wait time until a certain number of events occurs. Assume x1, . . . , xn ∼iid Gamma(α, β), where α is shape and β is rate. Calculate the maximum likelihood estimator of β either by hand or via R, but not both. The probability density function (PDF) for the Gamma distribution is: Calculate the log-likelihood function of the function below. I t doesnt have to be on r studi you can do by your handThe table below represents the probability density function for the random variable X. Find the standard deviation of X. Round the final answer to two decimal places. x P(X = x) 1 1/4 3 1/4 4 1/22. Consider a gas station's daily sales. Let Y be the volume of gas sold per day in 1000s of gallons. Assume the distribution of Y follows this pdf: 1sxs 2 S (v) . else What value of k is required for the above function to be a valid pdf? b. Verify that the function given is a valid probability density а. function c. Determine the cumulative distribution function (cdf) of the random variable d. Compute the probability that there is exactly 1500 gallons of sales on a given day (Hint: Find?(r - 1.5)) = 1 CO0
- Let X-N(2,5). Let f be the density of this random variable. We want to evaluate in R f(6). Which of the following statements are true? The normal distribution N(2, 5) 0.08 - 0.06- 0.04 - 0.02 - 0.00 -10 10 The answer is The R instruction is dnorm(-2,2,5) Can you explain why ? Density f(x)For a random variable which can be defined using the probability density function below, what is the probability that X is between 1.5 and 3.2? f(x) = {C(1+ x) ,1Let theta be the unknown market share that a company has for its primary product. Uncertainty in theta is represented by a probability density function of the beta distributional form with a = 54 and b = 46. You have developed the posterior distribution for theta after observing that 64 out of a random sample of 90 customers used the product. Which statement is correct? O A. In the sample, the proportion of people who used the product is greater than the expected proportion based on the prior, but this discrepancy has not affected the mean of the posterior distribution. B. In the sample, the proportion of people who used the product is less than the expected proportion based on the prior, and has led to the posterior distribution having a mean smaller than the prior mean but greater then the sample mean. O C. In the sample, the proportion of people who used the product is greater than the expected proportion based on the prior, and has led to the posterior distribution having a mean…Let X₁,..., Xn be a random sample from a continuous distribution with the probability density function fx(x; 0) = [3(x−0)², 0≤x≤0 +1, otherwise 0, Here, is an unknown parameter. Assume that the sample size n = 10 and the observed data are 1.46, 1.72, 1.54, 1.75, 1.77, 1.15, 1.60, 1.76, 1.62, 1.57 =For a random variable which can be defined using the probability density function below, what is the probability that X is between 1.3 and 3.1? с (1 + х) ,1 <A continous random variable has density function f(x)=(3x?)/(63) for Osxse, for some 0>0, and zero otherwise. What is the ratio of the mode to the median for this distribution? A. 2(1/3) B. 0/(21/3) O C.e D. 2(-1/3)The table below represents the probability density function for the random variable X. Find the standard deviation of X. x P(X = x) 0 1/6 1 1/6 4 1/3 7 1/3A density curve consists of the line segment connecting the points (0,1) and (0.5,1) and the segment connecting(0.5, 1) to the x-axis.a. Determine the coordinate point where the second segment crosses the x-axis.b. Determine the slope of that segmentc. Determine the equation of the line containing this segment (y = mx + b)d. Calculate the probability P(X > 1)Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. FreemanMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman