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##### Statistics 100A Homework Problems

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Question;1. In a quiz show, the contestant is given 4 choices for answer, only one of which iscorrect. When the contestant chooses an answer, the quiz master declares one of the remainingwrong options to be wrong, and asks the contestant if he would like to switch. Assuming hisinitial guess was random, is switching going to help?2. Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2orange balls. Suppose that we win $2 for each black ball selected and we lose $1 for eachwhite ball selected, no money is gained or lost for any orange balls picked. Let X denote ourwinnings.a. Find the pmf of X.b. Find E(X).c. Find V ar(X).3. A coin with probability p of head is tossed until either head or tail has occurredtwice. Find, in terms of p, the expected number of coin tosses needed.4. A fair 6-sided die is thrown 6 times. For i = 1, 2,:::, 6, let Xi = 1 if the i-th sideappears at least once in the 6 throws, and 0 otherwise.a. Find P(X1 = 0).b. Using your answer to part (a) or otherwise, find the number of different sidesthat we expect to see. Hint: How is this related to X1,:::, X6?5. A simplified model for the movement of the price of a stock supposes that on eachday the stock's price either moves up 1 unit with probability p or moves down 1 unit withprobability 1 p. The changes on different days are assumed to be independent.a. What is the probability that after 2 days the stock will be at its original price?b. What is the probability that after 3 days the stock's price will have increasedby 1 unit?c. Given that the after 3 days the stock's price has increased by 1 unit, what is the probability thatit went up on the 6. [12 points] Suppose that when in flight, airplane engines will fail withprobability 1 p independently from engine to engine. If an airplane needs a majority of its engines operative to make a successful flight, for what values of p is a 5-engine plane preferable to a 3-engineplane?7. Suppose that the number of earthquakes occurring in a certain region in Californiafollows a Poisson distribution at a rate of 7 per year.a. What is the probability of there being no earthquakes in one year?b. What is the probability that in exactly three of the next eight years no earthquakes will occur?8. Let X Geo(p), that is, P(X = x) = p(1.. p)x..1 for x = 1, 2,:::. Show thatP(X > y + xX > y) = P(X > x) for all x, y 1. This is known as the memoryless propertyof the geometric RV.9. Marc and Jacob play a series of card games until one of them wins five games.Suppose that the games are independent and the probability that Marc wins a game is 0.58.a. What is the probability that the series ends in seven games?b. If the series ends in seven games, what is the probability that Marc wins?10. A purchaser of a electrical components buys them in lots of size 10. It is his policy toinspect 3 components randomly from a lot and to accept the lot only if all 3 are nondefective.If 30% of the lots have 4 defective components and 70% have only 1 defective component,what proportion of the lots does the purchaser reject

Paper#61524 | Written in 18-Jul-2015

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