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STOCH - P ROBLEMS FOR THE MIDDLE TERM EXAM[1 Given three...

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P ROBLEMS FOR THE MIDDLE TERM EXAM [1] Given three sets A, B, C . Use the Venn diagram to illustrate the following sets: A B C , A B B ( C \ ( A B )) . [2] Let a card be selected from two ordinary packs of 52 cards. Denote A = { the card is Diamonds or clubs } , B = { the card is a red suit one } , C = { the card is not a Heart Jack } . Compute the following probabilities: P ( A ) , P ( B C ) , P ( C | B ) [3] EX 1( 2 points) : What are a sample space, events an their probilitiy? Formulate axioms of probability ? Give an example ? EX 2 ( 2 points) : What is the conditional probability of an event A given E . When three events A i , i = 1 , 2 , 3 are independent ? Give an example of a sequence of dependent ( non-independent) events A i , i 3 such that each pair of its membrs are independent. EX 3 (2 points) : A pair of fair dice is tossed. We obtain the finite equiprobable Ω consisting of the 36 ordered pairs of numbers between 1 and 6 Ω = { (1 , 1) , (1 , 2) , · · · , (6 , 6) } . Let X assign to each point ( a, b ) in Ω the minimum of its numbers, i.e X ( a, b ) = min( a, b ). What is the image set of random variable X ? Find the distribution of X and its expectation value, variance and standard deviaton. EX 3 ( 2 points) Let a pair of fair dice be tossed. If the sum is 5, find the probability that one of the dice is a 2. EX 4( 2 points) : Give a definition of a random variabe X on a sample space Ω and its distribution function and, emphasize the above concepts for the case that if Ω is a discrete space. EX 5( 2points) : A drawer contains red socks and white socks. When two socks are drawn at random, the probability that both socks are red and the third one is white is 1 2 . (i) How small can the number of socks in the drawer be ? (ii) If the number of white is divisible by 3, how small can the number of socks be ? 1
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EX 6 (2’) : A pair of fair dice is tossed. We obtain the finite equiprobable Ω consisting of the 36 ordered pairs of numbers between 1 and 6 Ω = { (1 , 1) , (1 , 2) , · · · , (6 , 6) } . Let X assign to each point ( a, b ) in Ω the minimum of its numbers, i.e X ( a, b ) = min( a, b ). What is the image set of random variable X ? Find the distribution of X and its expectation value, variance and standard deviaton. EX 7 : (4.20) Box A contains nine cards numbered 1through 9,and bx B contains five cards numbered 1 through 5. A box is chosen at random and a card drawn. if the number is even, find the probability that the card came from box A. EX 8 :(4.23) Let A be the event that a family has children of both sexes, and let B denote the event that a family has at most one boy. The events A and B are independent. How small can the number of children in a family be ? EX 9 Three machines A,B and C produce respectively 50%, 30% and 20% of total number of items of a factory. The percentages of defective output of these machines are respectively 2% , 3% and 4%. An item is selected at random and is found defective.
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