Math 230 February 2013 Midterm Exam 1 Question 3(20 pts).This question has two parts.a) Suppose that 13 cards are dealt without replacement from a standard deck of 52 cards.(i) What is the probability that exactly three of the cards will be spades?(ii) What is the probability that six cards will be of one suit, and the remaining seven willbe of another suit? 5
Math 230 February 2013 Midterm Exam 1 (iii) What is the probability that six cards will be of one suit, six cards will be of a 2nd suitand the remaining one card will be from a 3nd suit? (Notice how this is different fromthe previous question)b) Now suppose that 4 cards are dealt with replacement from a standard deck of 52 cards. Whatis the probability that there will be at least two jacks among the 4 cards? 6
Math 230 February 2013 Midterm Exam 1 Question 4(15 points).LetXhave meanaandYhave meanb. LetV ar(X) =σ2,E(Y2) =s2.a) For this part assume thatXandYare independent random variables. IfZ=13X−Y, findV ar(Z), andEXZ.b) If we now assume thatXandYare no longer independent random variable and thatc=E[XY]. If in additionY >0 then define the functionKbyK(m) =E[Y(X−m)2].(i) Find the value ofmwhich minimizesK(m).(ii) Why does your result give the expected answer ifXandYare now assumed to beindependent ? 7
Math 230 February 2013 Midterm Exam 1
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