Problem 2. If XN(0,1), determine the number of expected outcomes between 1 and 3 out of...
Problem 3. (Law of Large Number and Moving Average Model) Let s0, E1, E2, be a sequence of i.i.d. N(0,1) distributed random variable. Define a new sequence of random variables X1, X2, X3,-.. , as: | ; Xn = uEn + O€n-1; 1 Xi, answer the following ques- where and 0 are constant parameters. Define Xn _ =1 n tions: 1) Find out Var(Xn); 2) Show that X >u as n -> c0.
R CODE PROGRAM 1. Suppose we want to simulate an experiment that can take outcomes 1; : : : ; n with probabilies p1; : : : ; pn. To be specic, suppose the R-vector p=c(.1,.2,.3,.35, .02, .03) gives the desired probabilities. Write R code that produces a number from 1 to 6 with the given probabilities, without using if statements. I recommend using the R command cumsum to do this, though there many possible approaches. 2. Suppose we are...
Let Y be a continuous random variable having a gamma probability distribution with expected value 3/2 and variance 3/4. If you run an experiment that generates one-hundred values of Y , how many of these values would you expect to find in the interval [1, 5/2]?
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Problem 2 (0.5 x 3 = 1.5 point ) Simulate a sample of y1, ..., 4100 from a simple linear model Y = 1 + 2x + €, where € ~ N(0,62), and x is an arithmetic sequence from 1 to 100, with a step size of 1. Run set.seed(1) to set the seed of R's random number generator so that the simulation can be reproduced. • Make...
Question 17 Given the following data: Normal distribution Mean 3 Standard deviation -1 Determine number of samples out of 100 samples taken that fall within plus - minus 2 standard deviations
Assignment 2: Connection between Confidence Intervals and Sampling Distributions: The purpose of this activity is to help give you a better understanding of the underlying reasoning behind the interpretation of confidence intervals. In particular, you will gain a deeper understanding of why we say that we are “95% confidentthat the population mean is covered by the interval.” When the simulation loads you will see a normal-shaped distribution, which represents the sampling distribution of the mean (x-bar) for random samples of...
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Part 3. (13 polnts) SImulatlon of Gamma Random Varlables Background: When we use the probablity denslty function to find probabiltles for a random varlable, we are using the density function as a model. This Is a smooth curve, based on the shape of observed outcomes for the random varlable. The observed distribution will be rough and may not follow the model exactly. The probabillty density curve, or...
Problem 6 Five applicants for a job are ranked according to ability, with being the best. These rankings are unknown to an employer, who simply hires two applicants at random. What is the probability that this employer hires exactly one of the two best applicants? Problem 7 Five motors ( through 5) are available for use, and motor 2 is defective. Motors 1 and 2 come from supplier I, and motors 3, 4, and 5 come from supplierIL. Suppose two...
X = number of books Probability 1 0.05 2 0.10 3 0.20 4 0.35 5 0.10 6 0.15 7 0.05 13. For this problem, look at the "number of books" problem above. (a) Compute the expected value of X and interpret its meaning. Answer: E(X) = 4 (b) How many books are expected to be purchased if the enrollment is 20,000 students? Answer: Expected number of books = 4.20,000 = 80,000 (4.2.1)
7. Problem: (Fixed point iterations) Let f [0,3] [0,3] be a contraction with contraction constant q = 1 /3. Hence, f has a unique fixed point х* є [0,3]. n+1 f (xn), start: xo-0. Determine the smallest valid lower bound on the number of iterations n that are required to guarantee that x -n000
7. Problem: (Fixed point iterations) Let f [0,3] [0,3] be a contraction with contraction constant q = 1 /3. Hence, f has a unique fixed point...