document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164809>Question 1164809</A>: <I><SPAN STYLE=\"word-break: break-all;\"> let x1,x2,x3,x4 be a random sample of size 4 from a standard normal population.find the distribution of the statistics x1+x2/square root(x2 square +3x square) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853516 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=853516\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find the distribution of the statistic , we first need to evaluate the properties of the numerator and denominator based on the fact that  are i.i.d. .\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "### 1. Analyze the Numerator ()\r<BR>\n" );
document.write( "\n" );
document.write( "Let .<BR>\n" );
document.write( "Since  and  are independent normal variables:\r<BR>\n" );
document.write( "\n" );
document.write( "* **Mean:** <BR>\n" );
document.write( "* **Variance:** <BR>\n" );
document.write( "Therefore, . We can standardize this by noting that .\r<BR>\n" );
document.write( "\n" );
document.write( "### 2. Analyze the Denominator\r<BR>\n" );
document.write( "\n" );
document.write( "There appears to be a slight typo in your expression ( square). Assuming the statistic is meant to be , or a similar variation involving independent variables, let's look at the standard form of a **Student's t-distribution**.\r<BR>\n" );
document.write( "\n" );
document.write( "A t-distribution is formed by:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "where , , and  and  are independent.\r<BR>\n" );
document.write( "\n" );
document.write( "### 3. Re-evaluating the specific statistic\r<BR>\n" );
document.write( "\n" );
document.write( "If the statistic is exactly :\r<BR>\n" );
document.write( "\n" );
document.write( "1. **Numerator:** , where .<BR>\n" );
document.write( "2. **Denominator:**  is the sum of the squares of 2 independent standard normal variables, so .\r<BR>\n" );
document.write( "\n" );
document.write( "Substituting these into :\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divide both the numerator and denominator by  (the square root of the degrees of freedom ):\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "### 4. Conclusion\r<BR>\n" );
document.write( "\n" );
document.write( "By definition,  follows a **Student's t-distribution with 2 degrees of freedom**.\r<BR>\n" );
document.write( "\n" );
document.write( "**The distribution of the statistic is .**\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "### If the statistic was  specifically:\r<BR>\n" );
document.write( "\n" );
document.write( "This would be more complex because  appears in both the numerator and denominator, meaning they are **not independent**. In such cases, the distribution would not follow a standard t-distribution.\r<BR>\n" );
document.write( "\n" );
document.write( "However, in most textbook problems of this type, the variables in the denominator are independent of those in the numerator. Would you like me to show the derivation for a different combination of these variables? </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );