document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=252363>Question 252363</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me with this problem. I'm not really good at probability and statistics. A quick response would be nice. Thank you.\r<BR>\n" );
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document.write( "Given a PDF of variable x:<BR>\n" );
document.write( "      0    if x<-1<BR>\n" );
document.write( "f(x)= ax+a if -1=<x<0<BR>\n" );
document.write( "     -ax+a if 0=<x=<1<BR>\n" );
document.write( "      0    if x>1\r<BR>\n" );
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document.write( "a)Find a?<BR>\n" );
document.write( "b)Find the probability that the random variable X has values within the interval (1/2,1) and within the interval (-1/3,1/3)?<BR>\n" );
document.write( "c)Find P(X=1/2)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #184240 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=184240\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given a PDF of variable x:<BR>\n" );
document.write( "      0     if x<-1<BR>\n" );
document.write( "f(x)= ax+a  if -1<= x <0<BR>\n" );
document.write( "      -ax+a if 0<= x <=1<BR>\n" );
document.write( "      0     if x>1<BR>\n" );
document.write( "a)Find a?<BR>\n" );
document.write( "Solvd:<BR>\n" );
document.write( "Integral from -1 to 0 of (ax+a) + Integral from 0 to 1 of (-ax+a) = 1<BR>\n" );
document.write( "[(a/2)x^2+ax evaluated -1 to 0 + (-a/2)x^2+ax evaluated 0 to 1] = 1<BR>\n" );
document.write( "[(a/2)0 + a*0]-[(a/2)(-1)^2+a(-1)] + [(-a/2)(1^2)+a(1)]-[(-a/2)0+a*0]= 1<BR>\n" );
document.write( "-(a/2)+a + (-a/2)+a = 1<BR>\n" );
document.write( "a = 1<BR>\n" );
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document.write( "So now f(x) is defined as <BR>\n" );
document.write( "Given a PDF of variable x:<BR>\n" );
document.write( "      0     if x<-1<BR>\n" );
document.write( "f(x)= x+1  if -1<= x <0<BR>\n" );
document.write( "      -x+1 if 0<= x <=1<BR>\n" );
document.write( "      0     if x>1<BR>\n" );
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document.write( " <BR>\n" );
document.write( "b)Find the probability that the random variable X has values within the interval (1/2,1) and within the interval (-1/3,1/3)?<BR>\n" );
document.write( "P(1/2<x<1) = Integral (-x+1) eval btwn 1/2 and 1  <BR>\n" );
document.write( "= -(x^2/2)+x eval btwn 1/2 and 1<BR>\n" );
document.write( "= [-(1/2)+1]-[-(1/4)/2 + (1/2)]<BR>\n" );
document.write( "= [(1/2)]-[(-1/8)+(1/2)]<BR>\n" );
document.write( "= (1/2) - (3/8)<BR>\n" );
document.write( "= 1/8<BR>\n" );
document.write( "============================================<BR>\n" );
document.write( "P(-1/3 < x < 1/3) = <BR>\n" );
document.write( "Int(x+1)eval -1/3 to 0 + Int(-x+1) eval 0 to 1/3<BR>\n" );
document.write( "Then same procedure as above.<BR>\n" );
document.write( "Comment: Since these two areas are symmetric you<BR>\n" );
document.write( "could just evaluate one side, then double it.\r<BR>\n" );
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document.write( "c)Find P(X=1/2)? <BR>\n" );
document.write( "The probability of any particular value in a continuous<BR>\n" );
document.write( "distribution is always zero.<BR>\n" );
document.write( "Your problem would read Int(-x+1) eval from 1/2 to 1/2<BR>\n" );
document.write( "That is zero.<BR>\n" );
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document.write( "Cheers,<BR>\n" );
document.write( "Stan H. \r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );