1 Problem 1 True/False Section: For each statement indicate wheth...

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1 Problem 1 True/False Section: For each statement indicate whether it is always true or false. (a) If P(A) = P(Ac), then P(A) = 0.5. (b) Sum of the probabilities of all sample points in a sample space is less than or equal to one. (c) For any A and B, P(AUB) = P(A) + P(B) + P(AB). (d) For any A and B, P(AB) = P(A)P(B). (e) Each value of a CDF function is a probability density. (f) For any continuous random variable X, P(X = 3.5) is 0. 2 Problem 2 You have a coin, but you have no idea what are the chances it lands heads. The only thing you know that P(Heads) + P(Tails) = 1. You have identified three reasonable assumptions: p(Heads) = 0.5, p(Heads) = 0.3, p(Heads) = 0.1. Since you don't know much about coins, you assume that the chances that any of these assumptions is correct is 1/3. You proceed with 10 independent Bernoulli trials in which you toss a coin in the air and observe the outcome. You noticed that the coin landed 3 times heads and 7 times tails. (a) Draw a tree diagram that describes you experiment (first you decide which assumption is correct (1 out of 3), then you flip a coin and you either see 3 heads out of 10 flips or you don't. (b) What is the probability that you would observe 3 heads out of 10, considering the assumptions you've made. (c) What is the revised probability that the last assumption is correct (p(Heads) = 0.1) given the data you observed. 3 Problem 3 Your friend offers you to play a following card game. Before every round, a fair deck consisting of 52 cards is properly shuffled. Then you randomly select a card from the deck. If it is Jack or Queen, your friend pays you $15. If it is King or Ace, your friend pays you $5. If it is something else, you have to pay $4. Let X denote your winnings after one round. Hence, the possible values for X are: $15, $5, -$4. Note: You don't need to calculate the final answers exactly, i.e., your answer may look like 13*2+3-2 10 (no need to simplify). (a) Describe the pmf of X. (b) Find the expected value of X (c) If you play 10 rounds of this game, what are your expected winnings? 4 Problem 4 In order to get elected you need a support of at least 50% of population. You randomly select 100 voters for the exit-poll and ask them whether they voted for you. 40 voters out of 100 said that they voted for you. You decide to use the binomial test to check whether the probability that a random voter supports you is 0.5. (a) How many voters out of 100 you expect to be your supporters if 50% of population supports you? (b) What is the p-value in your binomial test? (c) How would you use the p-value from part (b) to decide whether the data (40 out of 100) is in line with your assumptions (50% support)?

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