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(MATH246)1998_midterm_exam.pdf
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MATH246,MidtermExamination,Fall1998

(1)(30pts):AcontinuousrandomvariableXhaspdf
.
C(1;x4);1.X.1fX(X). 0elsewhere
(a)FindC.
(b)FindthecdfofX.
(c)FindP[jXj.1].
2
(d)FindE[X]andE[X2]

(2)(30pts):Let(X.Y)havejointpdf
.
;y(1+x)
yex.0.y.0
fXY.
0otherwise
(a)FindthemarginalpdfofXandofY.
(b)AreX.Yindependent.
(c)FindtheprobabilityP(X.1.Y.1).
(d)FindtheprobabilityP(X+Y.2).

(2)(40pts):Thethreeballsnumbered1to3inanurnaredrawnatrandomoneatatimeuntil theurnisempty.Thesequenceoftheballnumbersisnoted
(a)Findthesamplespace. (b)FindthesetAkcorrespondingtotheevents\ballnumberkisselectedinthekthdraw" fork.1.2:
(c)AreA1.A2independent.
(d)FindP(A1\A2)
(e)De.nedtherandomvariableXtobethesummationofthenumbersonthe.rstandsecond balls.FindtheprobabilitymassfunctionforX.