document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183442>Question 1183442</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The lifetime of a mechanical assembly in a vibration test is exponentially<BR>\n" );
document.write( "distributed with a mean of 400 hours.<BR>\n" );
document.write( "(a) What is the probability that an assembly on test fails in less than 100 hours?<BR>\n" );
document.write( "(b) What is the probability that an assembly operates for more than 500 hours<BR>\n" );
document.write( "before failure?<BR>\n" );
document.write( "(c) If an assembly has been on test for 400 hours without a failure, what is the<BR>\n" );
document.write( "probability of a failure in the next 100 hours?<BR>\n" );
document.write( "(d) If 10 assemblies are tested, what is the probability that at least one fails in less<BR>\n" );
document.write( "than 100 hours? Assume that the assemblies fail independently.<BR>\n" );
document.write( "(e) If 10 assemblies are tested, what is the probability that all have failed by 800<BR>\n" );
document.write( "hours? Assume that the assemblies fail independently. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #813797 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=813797\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The pdf of the exponential distribution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+lambda%2Ae%5E%28-lambda%2Ax%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+lambda%2Ae%5E%28-lambda%2Ax%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=0 ALIGN=MIDDLE  ALT=\"x+%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and f(x) = 0 whenever x < 0.  <BR>\n" );
document.write( "The mean of the distribution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Flambda ALIGN=MIDDLE  ALT=\"1%2Flambda\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  <BR>\n" );
document.write( "The cdf is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F400%29%2Aint%28+e%5E%28-t%2F400%29%2C+dt%2C+0%2C+x%29+=+1-e%5E%28-x%2F400%29 ALIGN=MIDDLE  ALT=\"%281%2F400%29%2Aint%28+e%5E%28-t%2F400%29%2C+dt%2C+0%2C+x%29+=+1-e%5E%28-x%2F400%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "(a) What is the probability that an assembly on test fails in less than 100 hours?\r<BR>\n" );
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document.write( "Ans.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F400%29%2Aint%28e%5E%28-x%2F400%29%2C+dx%2C0%2C+100%29+=+1+-+e%5E%28-1%2F4%29+=+0.2212 ALIGN=MIDDLE  ALT=\"%281%2F400%29%2Aint%28e%5E%28-x%2F400%29%2C+dx%2C0%2C+100%29+=+1+-+e%5E%28-1%2F4%29+=+0.2212\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(b) What is the probability that an assembly operates for more than 500 hours<BR>\n" );
document.write( "before failure?\r<BR>\n" );
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document.write( "Ans. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F400%29%2Aint%28e%5E%28-x%2F400%29%2C+dx%2C500%2C+infinity%29+=+1-%281+-+e%5E%28-500%2F400%29%29+=+e%5E%28-5%2F4%29+=+0.2865 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(c) If an assembly has been on test for 400 hours without a failure, what is the probability of a failure in the next 100 hours?\r<BR>\n" );
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document.write( "Ans. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F400%29%2Aint%28e%5E%28-x%2F400%29%2C+dx%2C400%2C+500%29+=+e%5E%28-1%29+-+e%5E%28-5%2F4%29+=+0.0814 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "(d) If 10 assemblies are tested, what is the probability that at least one fails in less than 100 hours? Assume that the assemblies fail independently.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-%28%281%2F400%29%2Aint%28e%5E%28-x%2F400%29%2C+dx%2C100%2C+infinity%29%29%5E10+=+1-%28e%5E%28-1%2F4%29%29%5E10+=+0.9179 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(e) If 10 assemblies are tested, what is the probability that all have failed by 800<BR>\n" );
document.write( "hours? Assume that the assemblies fail independently.\r<BR>\n" );
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document.write( "Ans.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2F400%29%2Aint%28e%5E%28-x%2F400%29%2C+dx%2C0%2C+800%29%29%5E10+=+%281-e%5E%28-2%29%29%5E10+=+0.2336 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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