document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164808>Question 1164808</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 3 7. Let X1, X2, X3, X4 be a random sample of size 4 from a standard normal population. Find the sampling distribution (if possible) and moment gener- ating function of the statistic 2X?+3X2 +X +4X. What is the probability distribution of the sample? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853517 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=853517\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find the sampling distribution and the moment-generating function (MGF) of the given statistic, we rely on the properties of independent normal random variables.\r<BR>\n" );
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document.write( "### 1. The Probability Distribution of the Sample\r<BR>\n" );
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document.write( "Since  is a random sample of size 4 from a **standard normal population**, each  is independent and identically distributed (i.i.d.) such that:\r<BR>\n" );
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document.write( "The joint probability distribution of the sample is the product of their individual normal density functions:\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### 2. The Sampling Distribution of the Statistic\r<BR>\n" );
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document.write( "Let the statistic be .<BR>\n" );
document.write( "Note that for any , the square of the variable follows a **Chi-square distribution with 1 degree of freedom**:\r<BR>\n" );
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document.write( "However, our statistic  is a **weighted sum** of independent Chi-square variables:\r<BR>\n" );
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document.write( "Because the coefficients (weights) are not all equal to 1,  does **not** follow a standard Chi-square distribution. Instead, it follows a **Generalized Chi-square distribution** (specifically, a linear combination of Chi-squares). There is no simple named probability density function for this, but it is fully characterized by its MGF.\r<BR>\n" );
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document.write( "---\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 of a standard Chi-square variable  is:\r<BR>\n" );
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document.write( "Since the variables are independent, the MGF of a sum is the product of the individual MGFs. For a weighted variable , the MGF is .\r<BR>\n" );
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document.write( "Substitute the formula:\r<BR>\n" );
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document.write( "**Condition for Existence:** The MGF is defined only when all terms inside the square root are positive. The strictest constraint comes from , so .\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Summary\r<BR>\n" );
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document.write( "* **Sample Distribution:** Jointly Normal .<BR>\n" );
document.write( "* **Statistic Distribution:** A weighted sum of  variables (Generalized Chi-square).<BR>\n" );
document.write( "* **MGF:**  for .\r<BR>\n" );
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document.write( "Would you like me to calculate the **mean** and **variance** of this statistic using its MGF? </SPAN>\n" );
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