document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178271>Question 1178271</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that the number of cars arriving in 1 hour at a busy intersection is a Poisson probability distribution with λ = 100. Find, using Chebyshev’s inequality, a lower bound for the probability that the number of cars arriving at the intersection in 1 hour is between 70 and 130. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850354 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=850354\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem using Chebyshev's Inequality.\r<BR>\n" );
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document.write( "**Understanding the Problem**\r<BR>\n" );
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document.write( "* **Poisson Distribution:** The number of cars arriving in 1 hour follows a Poisson distribution with λ = 100.<BR>\n" );
document.write( "* **Mean and Variance:** For a Poisson distribution, the mean (μ) and variance (σ²) are both equal to λ. Therefore, μ = 100 and σ² = 100, so σ = √100 = 10.<BR>\n" );
document.write( "* **Interval:** We want to find a lower bound for P(70 ≤ X ≤ 130).\r<BR>\n" );
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document.write( "**Chebyshev's Inequality**\r<BR>\n" );
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document.write( "Chebyshev's Inequality states that for any random variable X with mean μ and standard deviation σ, and for any k > 0:\r<BR>\n" );
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document.write( "* P(|X - μ| ≥ kσ) ≤ 1/k²\r<BR>\n" );
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document.write( "We can also write it as:\r<BR>\n" );
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document.write( "* P(|X - μ| < kσ) ≥ 1 - 1/k²\r<BR>\n" );
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document.write( "**Applying Chebyshev's Inequality**\r<BR>\n" );
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document.write( "1.  **Find k:**<BR>\n" );
document.write( "    * We want to find P(70 ≤ X ≤ 130), which is equivalent to P(|X - 100| ≤ 30).<BR>\n" );
document.write( "    * We need to find k such that kσ = 30.<BR>\n" );
document.write( "    * Since σ = 10, we have k * 10 = 30, so k = 3.\r<BR>\n" );
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document.write( "2.  **Apply the Inequality:**<BR>\n" );
document.write( "    * P(|X - 100| < 30) ≥ 1 - 1/k²<BR>\n" );
document.write( "    * P(|X - 100| < 30) ≥ 1 - 1/3²<BR>\n" );
document.write( "    * P(|X - 100| < 30) ≥ 1 - 1/9<BR>\n" );
document.write( "    * P(|X - 100| < 30) ≥ 8/9\r<BR>\n" );
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document.write( "3.  **Interpret the Result:**<BR>\n" );
document.write( "    * P(70 ≤ X ≤ 130) ≥ 8/9<BR>\n" );
document.write( "    * 8/9 ≈ 0.8889\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "Using Chebyshev's Inequality, a lower bound for the probability that the number of cars arriving at the intersection in 1 hour is between 70 and 130 is 8/9 (approximately 0.8889).<BR>\n" );
document.write( " </SPAN>\n" );
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