document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171023>Question 1171023</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let X be the life span of one light bulb of a particular brand and Y be the lifespan of a bulb of another brand.  Suppose that X is N(965, 19^2) and Y is N(995, 31^2).  If one of each brand of bulb is randomly selected, Find P(X > Y). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851000 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851000\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**Given Information:**\r<BR>\n" );
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document.write( "* X ~ N(965, 19²)  (X is the lifespan of the first brand, normally distributed with mean 965 and variance 19²)<BR>\n" );
document.write( "* Y ~ N(995, 31²)  (Y is the lifespan of the second brand, normally distributed with mean 995 and variance 31²)\r<BR>\n" );
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document.write( "We want to find P(X > Y).\r<BR>\n" );
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document.write( "**1. Define a New Variable**\r<BR>\n" );
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document.write( "Let W = X - Y. Then, we need to find P(W > 0).\r<BR>\n" );
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document.write( "**2. Find the Mean and Variance of W**\r<BR>\n" );
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document.write( "* Mean of W (μ<sub>W</sub>) = E(W) = E(X - Y) = E(X) - E(Y) = 965 - 995 = -30<BR>\n" );
document.write( "* Variance of W (σ<sub>W</sub>²) = Var(W) = Var(X - Y) = Var(X) + Var(Y) (since X and Y are independent)<BR>\n" );
document.write( "    * σ<sub>W</sub>² = 19² + 31² = 361 + 961 = 1322<BR>\n" );
document.write( "* Standard deviation of W (σ<sub>W</sub>) = √1322 ≈ 36.359\r<BR>\n" );
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document.write( "Therefore, W ~ N(-30, 1322).\r<BR>\n" );
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document.write( "**3. Standardize W**\r<BR>\n" );
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document.write( "We need to find P(W > 0). To do this, we standardize W using the z-score formula:\r<BR>\n" );
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document.write( "* Z = (W - μ<sub>W</sub>) / σ<sub>W</sub><BR>\n" );
document.write( "* Z = (0 - (-30)) / √1322 = 30 / √1322 ≈ 30 / 36.359 ≈ 0.825\r<BR>\n" );
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document.write( "**4. Find the Probability**\r<BR>\n" );
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document.write( "We want to find P(W > 0), which is equivalent to P(Z > 0.825).\r<BR>\n" );
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document.write( "* P(Z > 0.825) = 1 - P(Z < 0.825)\r<BR>\n" );
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document.write( "Using a standard normal distribution table or a calculator, we find:\r<BR>\n" );
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document.write( "* P(Z < 0.825) ≈ 0.7953\r<BR>\n" );
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document.write( "Therefore:\r<BR>\n" );
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document.write( "* P(Z > 0.825) = 1 - 0.7953 = 0.2047\r<BR>\n" );
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document.write( "**Final Answer:**\r<BR>\n" );
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document.write( "P(X > Y) ≈ 0.2047<BR>\n" );
document.write( " </SPAN>\n" );
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