document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193590>Question 1193590</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose a fair coin is tossed three times. Let X be the random variable that represents the number of heads obtained. Find the cumulative distribution function of X. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848637 by </B><A HREF=/tutors/aboutme.mpl?userid=yurtman><B>yurtman(42)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=yurtman><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848637\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Determine the Probability Mass Function (PMF) of X**\r<BR>\n" );
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document.write( "* **Possible Values of X:** <BR>\n" );
document.write( "    * X = 0 (no heads): TTT <BR>\n" );
document.write( "    * X = 1 (one head): HTT, THT, TTH<BR>\n" );
document.write( "    * X = 2 (two heads): HHT, HTH, THH<BR>\n" );
document.write( "    * X = 3 (three heads): HHH\r<BR>\n" );
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document.write( "* **Probabilities:**<BR>\n" );
document.write( "    * P(X = 0) = 1/8 <BR>\n" );
document.write( "    * P(X = 1) = 3/8 <BR>\n" );
document.write( "    * P(X = 2) = 3/8 <BR>\n" );
document.write( "    * P(X = 3) = 1/8\r<BR>\n" );
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document.write( "**2. Define the Cumulative Distribution Function (CDF)**\r<BR>\n" );
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document.write( "* The CDF, denoted as F(x), gives the probability that the random variable X is less than or equal to a specific value (x).\r<BR>\n" );
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document.write( "**3. Calculate the CDF for Each Possible Value of X**\r<BR>\n" );
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document.write( "* **F(x) = P(X ≤ x)**\r<BR>\n" );
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document.write( "    * F(x) = 0 for x < 0 <BR>\n" );
document.write( "    * F(x) = P(X = 0) = 1/8 for 0 ≤ x < 1<BR>\n" );
document.write( "    * F(x) = P(X = 0) + P(X = 1) = 1/8 + 3/8 = 1/2 for 1 ≤ x < 2<BR>\n" );
document.write( "    * F(x) = P(X = 0) + P(X = 1) + P(X = 2) = 1/8 + 3/8 + 3/8 = 7/8 for 2 ≤ x < 3<BR>\n" );
document.write( "    * F(x) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 1/8 + 3/8 + 3/8 + 1/8 = 1 for x ≥ 3\r<BR>\n" );
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document.write( "**Therefore, the cumulative distribution function (CDF) of X is:**\r<BR>\n" );
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document.write( "* F(x) = 0 for x < 0<BR>\n" );
document.write( "* F(x) = 1/8 for 0 ≤ x < 1<BR>\n" );
document.write( "* F(x) = 1/2 for 1 ≤ x < 2<BR>\n" );
document.write( "* F(x) = 7/8 for 2 ≤ x < 3<BR>\n" );
document.write( "* F(x) = 1 for x ≥ 3<BR>\n" );
document.write( " </SPAN>\n" );
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