document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174870>Question 1174870</A>: <I><SPAN STYLE=\"word-break: break-all;\"> We have a random variable X = total number of success in n trials.<BR>\n" );
document.write( "If X is b(n,p) and assume n is some fixed number and p is to be estimated. Define the random variable p hat = X/n. Show that P hat is an unbiased estimator of p. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850630 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=850630\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down how to show that p̂ (p-hat) = X/n is an unbiased estimator of p.\r<BR>\n" );
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document.write( "**1. Definitions**\r<BR>\n" );
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document.write( "* **X:** The total number of successes in n trials.<BR>\n" );
document.write( "* **X ~ b(n, p):** X follows a binomial distribution with n trials and probability of success p.<BR>\n" );
document.write( "* **p̂ = X/n:** The estimator for p.<BR>\n" );
document.write( "* **Unbiased Estimator:** An estimator is unbiased if its expected value is equal to the true parameter it's estimating. In this case, we need to show E[p̂] = p.\r<BR>\n" );
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document.write( "**2. Expected Value of X**\r<BR>\n" );
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document.write( "* Since X ~ b(n, p), the expected value of X is:<BR>\n" );
document.write( "    * E[X] = np\r<BR>\n" );
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document.write( "**3. Expected Value of p̂**\r<BR>\n" );
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document.write( "* We want to find E[p̂] = E[X/n].<BR>\n" );
document.write( "* Using the linearity of expectation, we can write:<BR>\n" );
document.write( "    * E[p̂] = E[X/n] = (1/n) * E[X]\r<BR>\n" );
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document.write( "**4. Substitute E[X]**\r<BR>\n" );
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document.write( "* Substitute E[X] = np into the equation:<BR>\n" );
document.write( "    * E[p̂] = (1/n) * (np)\r<BR>\n" );
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document.write( "**5. Simplify**\r<BR>\n" );
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document.write( "* E[p̂] = (np)/n<BR>\n" );
document.write( "* E[p̂] = p\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "Since E[p̂] = p, we have shown that p̂ = X/n is an unbiased estimator of p. This means that, on average, the sample proportion (p̂) will equal the true population proportion (p).<BR>\n" );
document.write( " </SPAN>\n" );
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