document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=860902>Question 860902</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help. Any help would be appreciated. Thank you.\r<BR>\n" );
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document.write( "We throw one fair coin and one fair die (they are indipendent) and we define the Random Variables X and Y so that the following outcomes are described: X=1 if we get H(Heads), and X=0 if we get T(Tails) and Y=outcome of the die. We also define the Random Variable Z = XY.<BR>\n" );
document.write( "A) Define the sample space S and the Random Variables X and Y as functions of S.<BR>\n" );
document.write( "B) Find the set of Rnadom Variable Z and count its mass of probability.<BR>\n" );
document.write( "C) Find the mean value (E(Z)) and dispersion (Var(Z)) of Random Variable Z. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #518778 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=518778\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A)Student says that the die is a 6 sided die <BR>\n" );
document.write( "our sample space consists of pairs (a, b) where a is 0 or 1 depending on the coin toss (1 = heads, 0 = tails) and b is 1, 2, 3, 4, 5, 6 depending on die toss, we assume the die has 6 sides and each side is numbered<BR>\n" );
document.write( "here is the sample space <BR>\n" );
document.write( "(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)<BR>\n" );
document.write( "(0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (0, 6)<BR>\n" );
document.write( "there are 12 possible outcomes when throwing the coin and dice<BR>\n" );
document.write( "S(X) is 1 if coin toss is heads<BR>\n" );
document.write( "S(X) is 0 if coin toss is tails<BR>\n" );
document.write( "S(Y) is i if die toss is i, where i is 1, 2, 3, 4 5, 6<BR>\n" );
document.write( "B) sample space for Z when Z = XY<BR>\n" );
document.write( "Z value    probability<BR>\n" );
document.write( "  1           1/12<BR>\n" );
document.write( "  2           1/12<BR>\n" );
document.write( "  3           1/12<BR>\n" );
document.write( "  4           1/12<BR>\n" );
document.write( "  5           1/12<BR>\n" );
document.write( "  6           1/12<BR>\n" );
document.write( "  0           6/12 = 1/2<BR>\n" );
document.write( "adding up the probabilities we get 1/2 + 1/2 = 1<BR>\n" );
document.write( "C) E(z) = 0*(1/2) + 1*(1/12) + 2*(1/12) + ...+6*(1/12)<BR>\n" );
document.write( "   E(z) = 0 + (1/12)*(1 + 2 + 3 + 4 + 5 + 6)<BR>\n" );
document.write( "   E(z) = (1/12) * 21 = 21/12 = 7/4 = 1.75<BR>\n" );
document.write( "Var(Z) = (0 - 21/12)^2 * (1/2) + (1 - 21/12)^2 * (1/12) + (2 - 21/12)^2 * (1/12) + ... + (6 - 21/12)^2 * (1/12)<BR>\n" );
document.write( "Var(Z) = (0 - 1.75)^2 * (1/2) + (1/12) * ( (1 - 1.75)^2 + (2 - 1.75)^2 + (3 - 1.75)^2 + (4 - 1.75)^2 + (5 - 1.75)^2 + (6 - 1.75)^2 )<BR>\n" );
document.write( "Var(Z) = 1.53125 + (1/12) * 35.875 = 4.52\r<BR>\n" );
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document.write( " </SPAN>\n" );
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