document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177295>Question 1177295</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let X be a random variable with pdf f (x) = kx^2 where 0 ≤ x ≤ 1.<BR>\n" );
document.write( "(a) Find k.<BR>\n" );
document.write( "(b) Find E(X) and Var(X).<BR>\n" );
document.write( "(c) Find MX(t). Using the mgf, find E(X).\r<BR>\n" );
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document.write( "Thank You </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850478 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=850478\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **(a) Find k**\r<BR>\n" );
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document.write( "Since f(x) is a probability density function (pdf), the integral of f(x) over its domain must equal 1:\r<BR>\n" );
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document.write( "* ∫[0, 1] kx² dx = 1\r<BR>\n" );
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document.write( "Integrate:\r<BR>\n" );
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document.write( "* [ (kx³/3) ] from 0 to 1 = 1<BR>\n" );
document.write( "* (k/3) - (0) = 1<BR>\n" );
document.write( "* k/3 = 1<BR>\n" );
document.write( "* k = 3\r<BR>\n" );
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document.write( "Therefore, f(x) = 3x² for 0 ≤ x ≤ 1.\r<BR>\n" );
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document.write( "**(b) Find E(X) and Var(X)**\r<BR>\n" );
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document.write( "1.  **Find E(X):**\r<BR>\n" );
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document.write( "    * E(X) = ∫[0, 1] x * f(x) dx<BR>\n" );
document.write( "    * E(X) = ∫[0, 1] x * (3x²) dx<BR>\n" );
document.write( "    * E(X) = ∫[0, 1] 3x³ dx<BR>\n" );
document.write( "    * E(X) = [ (3x⁴/4) ] from 0 to 1<BR>\n" );
document.write( "    * E(X) = (3/4) - (0)<BR>\n" );
document.write( "    * E(X) = 3/4\r<BR>\n" );
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document.write( "2.  **Find E(X²):**\r<BR>\n" );
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document.write( "    * E(X²) = ∫[0, 1] x² * f(x) dx<BR>\n" );
document.write( "    * E(X²) = ∫[0, 1] x² * (3x²) dx<BR>\n" );
document.write( "    * E(X²) = ∫[0, 1] 3x⁴ dx<BR>\n" );
document.write( "    * E(X²) = [ (3x⁵/5) ] from 0 to 1<BR>\n" );
document.write( "    * E(X²) = (3/5) - (0)<BR>\n" );
document.write( "    * E(X²) = 3/5\r<BR>\n" );
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document.write( "3.  **Find Var(X):**\r<BR>\n" );
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document.write( "    * Var(X) = E(X²) - [E(X)]²<BR>\n" );
document.write( "    * Var(X) = 3/5 - (3/4)²<BR>\n" );
document.write( "    * Var(X) = 3/5 - 9/16<BR>\n" );
document.write( "    * Var(X) = (48 - 45) / 80<BR>\n" );
document.write( "    * Var(X) = 3/80\r<BR>\n" );
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document.write( "**(c) Find MX(t) and Use the mgf to Find E(X)**\r<BR>\n" );
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document.write( "1.  **Find MX(t) (Moment Generating Function):**\r<BR>\n" );
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document.write( "    * MX(t) = E(e^(tX)) = ∫[0, 1] e^(tx) * f(x) dx<BR>\n" );
document.write( "    * MX(t) = ∫[0, 1] e^(tx) * (3x²) dx\r<BR>\n" );
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document.write( "    We use integration by parts twice. Let u = 3x², dv = e^(tx) dx.<BR>\n" );
document.write( "    Then du = 6x dx, v = e^(tx)/t.\r<BR>\n" );
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document.write( "    * MX(t) = [ 3x²e^(tx)/t ] from 0 to 1 - ∫[0, 1] 6xe^(tx)/t dx<BR>\n" );
document.write( "    * MX(t) = 3e^t/t - (6/t) ∫[0, 1] xe^(tx) dx\r<BR>\n" );
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document.write( "    Now, integrate ∫xe^(tx) dx by parts again. Let u = x, dv = e^(tx) dx.<BR>\n" );
document.write( "    Then du = dx, v = e^(tx)/t.\r<BR>\n" );
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document.write( "    * ∫xe^(tx) dx = [ xe^(tx)/t ] from 0 to 1 - ∫[0, 1] e^(tx)/t dx<BR>\n" );
document.write( "    * ∫xe^(tx) dx = e^t/t - [ e^(tx)/t² ] from 0 to 1<BR>\n" );
document.write( "    * ∫xe^(tx) dx = e^t/t - (e^t/t² - 1/t²)<BR>\n" );
document.write( "    * ∫xe^(tx) dx = e^t/t - e^t/t² + 1/t²\r<BR>\n" );
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document.write( "    Substitute back into MX(t):\r<BR>\n" );
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document.write( "    * MX(t) = 3e^t/t - (6/t) [ e^t/t - e^t/t² + 1/t² ]<BR>\n" );
document.write( "    * MX(t) = 3e^t/t - 6e^t/t² + 6e^t/t³ - 6/t³<BR>\n" );
document.write( "    * MX(t) = (3te^t - 6e^t + 6e^t/t - 6/t²)/t<BR>\n" );
document.write( "    * MX(t) = (3t^2e^t - 6te^t + 6e^t - 6)/t^3\r<BR>\n" );
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document.write( "2.  **Find E(X) using MX(t):**\r<BR>\n" );
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document.write( "    * E(X) = MX'(0)<BR>\n" );
document.write( "    * MX'(t) = d/dt [(3t²e^t - 6te^t + 6e^t - 6)/t^3]\r<BR>\n" );
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document.write( "    We can use L'Hopital's rule or series expansions to find the limit.\r<BR>\n" );
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document.write( "    * E(X) = lim(t→0) MX'(t)<BR>\n" );
document.write( "    * Using L'Hopital's rule three times, or the series expansion of e^t, we find:<BR>\n" );
document.write( "    * E(X) = lim(t→0) (3t^2e^t+6te^t+3e^t-6e^t-6te^t+6e^t)/3t^2<BR>\n" );
document.write( "    * E(X) = lim(t→0) (3t^2e^t+3e^t)/3t^2<BR>\n" );
document.write( "    * E(X) = lim(t→0) (3t^2e^t+6te^t+6e^t)/6t<BR>\n" );
document.write( "    * E(X) = lim(t→0) (3t^2e^t+12te^t+6e^t)/6<BR>\n" );
document.write( "    * E(X) = 6/8 = 3/4\r<BR>\n" );
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document.write( "**Answers**\r<BR>\n" );
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document.write( "* **(a) k = 3**<BR>\n" );
document.write( "* **(b) E(X) = 3/4, Var(X) = 3/80**<BR>\n" );
document.write( "* **(c) MX(t) = (3t²e^t - 6te^t + 6e^t - 6)/t^3, E(X) = 3/4**<BR>\n" );
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