Stat350-Rev-exam - DECEMBER EXAMINATIONS STATISTICS 350 SECTION[F01 Question 1[4 marks A pocket contains three coins one of which has heads on both

Stat350-Rev-exam - DECEMBER EXAMINATIONS STATISTICS 350...

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DECEMBER EXAMINATIONS STATISTICS 350, SECTION [F01] Question 1. [4 marks]A pocket contains three coins, one of which has heads on both sides, whilethe other two are normal coins. A coin is chosen at random from the pocket and tossed three times.[1](a) Find the probability of obtaining three heads.[1](b) If each of the three tosses results in a head, what is the probability that the two-headed coinwas chosen?[2](c) What is the expected number of tosses resulting in a head?Question 2. [5 marks]In a game of marbles a shooter may either(A) miss (with probabilityp1= 0.4);(B) hit one marble out and stick in the ring (with probabilityp2= 0.2); or(C) hit one marble out and leave the ring (with probabilityp3= 0.4).If B occurs the shooter shoots again, but otherwise the turn ends. LetXbe a random variabledenoting the number of marbles hit out in a single turn.[2](a) Find the probabilty mass function (pmf)f(x)ofX.[1](b) What is the most probable (modal) value ofX?LetYbe a random variable denoting the number of shots in a turn.[1](c) Find the pmf ofY.[1](d) Find the conditional pmf ofXgivenY= 3.Question 3. [6 marks]Suppose thatXis a continuous random variable with cumulative distri-bution function (cdf)F(x) = 1eλxforx >0and zero otherwise.

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