document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192148>Question 1192148</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose the probability density of X is given by  <BR>\n" );
document.write( "f(x) = kxe^(-x^2 ), X>0<BR>\n" );
document.write( "       = 0, otherwise\r<BR>\n" );
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document.write( "(a) Find the value of k. <BR>\n" );
document.write( "(b) Find the distribution function of X, i.e., the cumulative density function of X.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849033 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=849033\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Find the value of k**\r<BR>\n" );
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document.write( "* **Condition for a valid probability density function (PDF):** <BR>\n" );
document.write( "    * The total area under the PDF curve must be equal to 1. <BR>\n" );
document.write( "    * Mathematically, this means: ∫[from 0 to ∞] f(x) dx = 1\r<BR>\n" );
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document.write( "* **Set up the integral:**<BR>\n" );
document.write( "    * ∫[from 0 to ∞] kxe^(-x^2) dx = 1\r<BR>\n" );
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document.write( "* **Solve the integral:**<BR>\n" );
document.write( "    * Let u = -x^2 <BR>\n" );
document.write( "    * du = -2x dx <BR>\n" );
document.write( "    * dx = -du / (2x)\r<BR>\n" );
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document.write( "    * Substitute: ∫[from 0 to ∞] kxe^u * (-du / (2x)) <BR>\n" );
document.write( "                      = -k/2 ∫[from 0 to -∞] e^u du <BR>\n" );
document.write( "                      = -k/2 [e^u] [from 0 to -∞]<BR>\n" );
document.write( "                      = -k/2 [0 - 1] <BR>\n" );
document.write( "                      = k/2\r<BR>\n" );
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document.write( "* **Solve for k:**<BR>\n" );
document.write( "    * k/2 = 1<BR>\n" );
document.write( "    * k = 2\r<BR>\n" );
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document.write( "**Therefore, the value of k is 2.**\r<BR>\n" );
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document.write( "**b) Find the distribution function of X (cumulative density function)**\r<BR>\n" );
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document.write( "* **Definition:** The cumulative distribution function (CDF), F(x), gives the probability that the random variable X takes on a value less than or equal to x.\r<BR>\n" );
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document.write( "* **Calculate F(x):**<BR>\n" );
document.write( "    * For x ≤ 0: F(x) = P(X ≤ x) = 0 (since f(x) = 0 for x ≤ 0)<BR>\n" );
document.write( "    * For x > 0: <BR>\n" );
document.write( "        * F(x) = P(X ≤ x) = ∫[from 0 to x] f(t) dt <BR>\n" );
document.write( "                    = ∫[from 0 to x] 2te^(-t^2) dt \r<BR>\n" );
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document.write( "        * Let u = -t^2 <BR>\n" );
document.write( "        * du = -2t dt <BR>\n" );
document.write( "        * dt = -du / (2t)\r<BR>\n" );
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document.write( "        * Substitute: ∫[from 0 to -x^2] 2te^u * (-du / (2t)) <BR>\n" );
document.write( "                          = -∫[from 0 to -x^2] e^u du<BR>\n" );
document.write( "                          = -[e^u] [from 0 to -x^2]<BR>\n" );
document.write( "                          = -(e^(-x^2) - e^0)<BR>\n" );
document.write( "                          = 1 - e^(-x^2)\r<BR>\n" );
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document.write( "* **Combine the results:**\r<BR>\n" );
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document.write( "   * F(x) = <BR>\n" );
document.write( "       * 0, for x ≤ 0<BR>\n" );
document.write( "       * 1 - e^(-x^2), for x > 0\r<BR>\n" );
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document.write( "**Therefore, the distribution function of X is:**\r<BR>\n" );
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document.write( "   F(x) = { <BR>\n" );
document.write( "       0, for x ≤ 0<BR>\n" );
document.write( "       1 - e^(-x^2), for x > 0 <BR>\n" );
document.write( "   }<BR>\n" );
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