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# Monrad.410.13 - STAT 410/MATH 464 Name March 2 2006 BMW...

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Unformatted text preview: STAT 410/MATH 464 Name March 2, 2006 BMW Monrad MIDTERM EXAM OPEN BOOK SHOW ALL WORK Problem 1 160 points) Let X and Y be independent random variables, each geometrically distributed with parameter p: P(X=z) = P(Y=z) = p(l—p)z, for z=0, l, 2, 3, . (a) Find the joint pmf f(X, ) of X and Y. (b) Find P(X=Y). (c) Find P(X>Y). (d) Find the pmf g(z) = P(X+Y=z), for z = O, l, 2, (e) Find P(Y=y I X + Y = z), for y= 0,1, ..., z. (t) Compute E( Y I X + Y =2). Emblem 2 130 points) Let X and Y be independent random variables, each geometrically distributed with parameter p. Set Z = Y— X and M: min(X,Y). (a) Show that for integers z 2 0 and m 2 0, P (M=m, Z=z) = p2 (1-p>2m (1-p>z . Hint: If 2 Z 0, then (M=rn, Z=z) = (Xm, Y=m+z). (b) Show that for integers z<0 and m2 0, P <M=m, Z=z> = p2 (1—12)“ (1W. (0) Show that M and Z are independent. Problem 3 140 points} Let X1 and X2 have joint p.d.f. ﬂxl, X2) = Ze ‘(X'Hz ), for 0< x1 <x2. (a) Are X1 and X2 independent? Explain. (b) Find the joint p.d.f. g(y1, y2) of the variables Y1:X2_X'1 and Y2=)(1 (c) Find the marginal p.d.f’s g1(yl) and g2(y2) of Y1 and Y2. (d) Are Y1 and Y2 independent? Explain. Problem 4 140 points! Let X; and X2 havejoint p.d.f. ﬂxl, x2) = 8x1x2, for 0<x1 <x2 <1. (a) Are X1 and X2 independent? Explain. (b) Find the joint p.d.f. g(yl, y2) of the variables Y1=X2 and Yszz/Xl. (c) Find the marginal p.d.f.’s g1 0/1) and g2 022) of Y1 and Y2. (d) Are Y1 and Y2 independent? Explain. Ditlev Monrad STAT 4lO/MATH 464 October 2, 2007 MIDTERM EXAM OPEN BOOK SHOW ALL WORK Problem 1. (25 points) Let the continuous variables X1, X2, X3 have the joint pdf ﬂx1,x2,x3) =6 for0<x1<xz<X3 < l. (a) Find the marginal pdfs f1(x1), fz(xz) and f3(x3) . (b) Are X1, X2, and X3 independent? Explain. Problem 2. (30 points) The continuous random variables X1 and X2 are independent and have marginal pdfs f1(x1) = l/xlz, x1 > 1, andfz (xz) = l/xf, x2 > 1, respectively. (a) Find the joint pdfﬂxl, x2) of X1 and X2. (b) Put W = XIX; and compute the cdf Fw(t)=P(WSt) , -00<t<00. (c) Determine the pdf of W. Problem 3. (50 points) Let X1 and X2 be continuous random variables with joint pdf f(x1,x2) =2(1+x1+x2)'3, for x1 >0, x2>0. (a) Find the marginal pdfs f1(x1) and f2(x2) of X1 and X2. (b) Are X1 and X2 independent? Explain. (c) Find the joint pdfg(yl, y2) of the variables X2 Y=X+X andY= . 1 1 2 2 X1+X2 (d) Find the marginal pdfs g1(yl) and g2(y2) of Y1 and Y2. (e) Are Y1 and Y2 independent? Explain. Problem 4. (60 points) Let X and Y be independent random variables, each geometrically distributed with parameter p: P(X=z) =P(Y=z) =p(l —p)z, forz= 0,1, 2, 3, (a) Find the joint pmf of X and Y, p(x, ) = P(X = X and Y = y). (b) Find P(X = Y). (c) Find P(X>Y). ((1) Find the pmfg(z) = P(X + Y = z), for z = O, l, 2, (e) FindP(Y=y lX+Y=z), fory=0, 1, ...,z. (0 Compute E( Y 1 X + Y = z). ...
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Monrad.410.13 - STAT 410/MATH 464 Name March 2 2006 BMW...

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