document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174433>Question 1174433</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A production line of 5000 transistors has an unknown number D of defectives. A sample of 100 is taken and we ascertain that this sample of transistors has 10 defectives. The maximum likelihood estimate for D is the number of defectives which gives the highest probability for obtaining the number of defectives observed in the sample. Guess this number D. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850659 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=850659\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let $N = 5000$ be the total number of transistors.<BR>\n" );
document.write( "Let $D$ be the unknown number of defective transistors.<BR>\n" );
document.write( "We take a sample of $n = 100$ transistors.<BR>\n" );
document.write( "We observe $d = 10$ defectives in the sample.\r<BR>\n" );
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document.write( "We want to find the maximum likelihood estimate for $D$.<BR>\n" );
document.write( "The probability of observing $d$ defectives in a sample of $n$ is given by the hypergeometric distribution:<BR>\n" );
document.write( "$$P(X=d) = \frac{\binom{D}{d} \binom{N-D}{n-d}}{\binom{N}{n}}$$<BR>\n" );
document.write( "We want to find the value of $D$ that maximizes this probability.\r<BR>\n" );
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document.write( "We can approximate the hypergeometric distribution with the binomial distribution when $n$ is much smaller than $N$.<BR>\n" );
document.write( "In our case, $n=100$ and $N=5000$, so $n/N = 100/5000 = 1/50 = 0.02$, which is small.<BR>\n" );
document.write( "So we can use the binomial approximation.\r<BR>\n" );
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document.write( "The probability of a transistor being defective is $p = D/N$.<BR>\n" );
document.write( "The probability of observing $d$ defectives in a sample of $n$ is:<BR>\n" );
document.write( "$$P(X=d) = \binom{n}{d} p^d (1-p)^{n-d}$$<BR>\n" );
document.write( "We want to maximize this probability with respect to $D$.<BR>\n" );
document.write( "Since $p = D/N$, we have:<BR>\n" );
document.write( "$$P(X=d) = \binom{n}{d} \left(\frac{D}{N}\right)^d \left(1-\frac{D}{N}\right)^{n-d}$$<BR>\n" );
document.write( "To find the maximum likelihood estimate, we can take the derivative of the log-likelihood function with respect to $D$ and set it to zero.<BR>\n" );
document.write( "The log-likelihood function is:<BR>\n" );
document.write( "$$L(D) = \ln\left(\binom{n}{d}\right) + d\ln\left(\frac{D}{N}\right) + (n-d)\ln\left(1-\frac{D}{N}\right)$$<BR>\n" );
document.write( "Taking the derivative with respect to $D$:<BR>\n" );
document.write( "$$\frac{dL}{dD} = \frac{d}{D} - \frac{n-d}{N-D}$$<BR>\n" );
document.write( "Setting the derivative to zero:<BR>\n" );
document.write( "$$\frac{d}{D} = \frac{n-d}{N-D}$$<BR>\n" );
document.write( "$$d(N-D) = D(n-d)$$<BR>\n" );
document.write( "$$dN - dD = nD - dD$$<BR>\n" );
document.write( "$$dN = nD$$<BR>\n" );
document.write( "$$D = \frac{dN}{n}$$<BR>\n" );
document.write( "Plugging in the given values:<BR>\n" );
document.write( "$$D = \frac{10 \times 5000}{100}$$<BR>\n" );
document.write( "$$D = \frac{50000}{100}$$<BR>\n" );
document.write( "$$D = 500$$\r<BR>\n" );
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document.write( "Therefore, the maximum likelihood estimate for $D$ is 500.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{500}$<BR>\n" );
document.write( " </SPAN>\n" );
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