document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164708>Question 1164708</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let X1, X2, X3, ...,X 50 be a random sample of size 50 from a distribution<BR>\n" );
document.write( "with density function given by:<BR>\n" );
document.write( "f(x) = product of x^(a-1) and exp^(x/Q) divided by product of gamma of alpha<BR>\n" );
document.write( "and Q^(a) with the condition 0 < x < inf. [Here a means alpha] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853524 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=853524\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To solve this problem, we first identify the distribution described. The density function you provided is:\r<BR>\n" );
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document.write( "This is the probability density function of a **Gamma Distribution**, denoted as .\r<BR>\n" );
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document.write( "### 1. Key Properties of the Distribution\r<BR>\n" );
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document.write( "For a single random variable  following this distribution, the mean and variance are defined as:\r<BR>\n" );
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document.write( "* **Mean ():** <BR>\n" );
document.write( "* **Variance ():** \r<BR>\n" );
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document.write( "### 2. Properties of the Random Sample\r<BR>\n" );
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document.write( "Since you have a random sample of size  (), we can determine the distribution of the **sample sum** and the **sample mean**.\r<BR>\n" );
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document.write( "#### A. Distribution of the Sum ()\r<BR>\n" );
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document.write( "The sum of independent Gamma variables with the same scale parameter  also follows a Gamma distribution.\r<BR>\n" );
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document.write( "* **New Alpha:** <BR>\n" );
document.write( "* **Scale:**  remains the same.\r<BR>\n" );
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document.write( "#### B. Distribution of the Sample Mean ()\r<BR>\n" );
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document.write( "By the **Central Limit Theorem**, since your sample size () is sufficiently large (typically ), the sampling distribution of the mean will be approximately **Normal**, regardless of the underlying Gamma shape.\r<BR>\n" );
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document.write( "The parameters for this normal distribution would be:\r<BR>\n" );
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document.write( "* **Mean of :** <BR>\n" );
document.write( "* **Variance of :** \r<BR>\n" );
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document.write( "### 3. Moment Generating Function (MGF)\r<BR>\n" );
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document.write( "The MGF for a single  is:\r<BR>\n" );
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document.write( "For the entire sample sum :\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "**Would you like me to calculate a specific probability for this sample, or find the maximum likelihood estimators (MLE) for  and ?** </SPAN>\n" );
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