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# The value distribution of paintings at a gallery is approx mount or bell shaped with mean and standard deviation

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ty distribution: x 3 5 8 10 pX(x) k k k 2k i. Determine the constant k and hence write down the probability distribution of X. ii. Find E(X), the expected value of X. iii. Find Var(X), the variance of X.
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y aces or twos, you lose the game immediately. You also lose if you draw picture cards(J,Q,K) more than twice. In this question, you’ll study the probability of winning this game.(a) What is the probability of drawing no aces or twos after thirteen draws?(b) Given you have drawn thirteen times, none of which is aces or twos, what is the probability that you draw at most two picture cards?(c) What is the probability to win this game? 12. Suppose you are tossing an unbiased coin for100times.(a) What is the probability of getting50heads and50tails?(b) LetXbe the random variable counting the number of heads you observe in this exper-iment. What is the expected value ofX? What is the variance ofX? What is thestandard deviation ofX? 13. The following are probability distributions for two random variablesX,Y. kPr(X=k) 0,0.4 1,0.3 2,0.3 kPr(Y=k) 0,0.5 1,0.3 2,0.2 (a) Construct the probability distribution table for the random variableXY.(b) Find E[X],E[Y] and E[XY]. Is is true that E[XY] =E[X]E[Y]?(c) Find the variances σ2X,σ2Y,σ2XY of X,Y and XY. Is it true that σ2XY=σ2Xσ2Y? 14. The aliens who are fond of gambling came back to play another game with you. In this game, you first toss a coin5times. If you observe3or fewer tails, you roll a die3times. If youobserve4or more tails, you roll a die20times. What is the probability that you end up with at most two6’s in your dice rolls? 15. (Challenge question, worth2points) You have two bags, each of which contains10marbles.Each time you remove a marble from a random bag. What is the probability that after one of the bags is emptied, there are still exactly3marbles in the other bag?
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- 1); I{x>1}, theta > 1. (a) Show that log Xi has an exponential distribution with a mean of 1/theta. (b) Find the form for a UMP test of H_0: theta <= theta_0 vs. H_a : theta > theta_0. (c) Give formula for nding the rejection region for a given value of alpha. Hint: use the result from (a) to fi nd the distribution of the test statistic. (d) Conduct the test in (b) with alpha = 0.05 and theta_0 = 1:5 using the dataset: {1.2, 2.4, 1.3, 1.7, 1.9}. (e) Find the form of a UMPU test for testing H_0 : theta = theta_0 vs. H_a : theta_0 != theta_0. (f) Use the data in (d) to conduct the test in (e) with alpha = 0.05 and theta_0 = 1.5.
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- 1); I{x>1}, theta > 1. (a) Show that log Xi has an exponential distribution with a mean of 1/theta. (b) Find the form for a UMP test of H_0: theta <= theta_0 vs. H_a : theta > theta_0. (c) Give formula for nding the rejection region for a given value of alpha. Hint: use the result from (a) to fi nd the distribution of the test statistic. (d) Conduct the test in (b) with alpha = 0.05 and theta_0 = 1:5 using the dataset: {1.2, 2.4, 1.3, 1.7, 1.9}. (e) Find the form of a UMPU test for testing H_0 : theta = theta_0 vs. H_a : theta_0 != theta_0. (f) Use the data in (d) to conduct the test in (e) with alpha = 0.05 and theta_0 = 1.5.
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- 1); I{x>1}, theta > 1. (a) Show that log Xi has an exponential distribution with a mean of 1/theta. (b) Find the form for a UMP test of H_0: theta <= theta_0 vs. H_a : theta > theta_0. (c) Give formula for nding the rejection region for a given value of alpha. Hint: use the result from (a) to fi nd the distribution of the test statistic. (d) Conduct the test in (b) with alpha = 0.05 and theta_0 = 1:5 using the dataset: {1.2, 2.4, 1.3, 1.7, 1.9}. (e) Find the form of a UMPU test for testing H_0 : theta = theta_0 vs. H_a : theta_0 != theta_0. (f) Use the data in (d) to conduct the test in (e) with alpha = 0.05 and theta_0 = 1.5.
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