document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174436>Question 1174436</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You do a following experiment: You toss a biased coin, with 1/3 probability for heads and 2/3 for tails. If the coin falls heads you choose uniformly from {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, if it turns out tails you choose randomly from {-1, -2, -3, -4, -5, -6}. Let X denote the random variable denoting your final outcome in this two stage experiment. Find variance of X. (Give only the variance of X as an answer, 2 decimals is enough, no other text!) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850658 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=850658\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely! Let's break down the calculation of the variance of X step by step.\r<BR>\n" );
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document.write( "**1. Define the Random Variables and Probabilities**\r<BR>\n" );
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document.write( "* **Coin Toss:**<BR>\n" );
document.write( "    * Let H be the event of getting heads, with P(H) = 1/3.<BR>\n" );
document.write( "    * Let T be the event of getting tails, with P(T) = 2/3.<BR>\n" );
document.write( "* **Choice of Numbers:**<BR>\n" );
document.write( "    * If heads (H), we choose a number from {1, 2, ..., 10} uniformly. Let's call this random variable Y.<BR>\n" );
document.write( "    * If tails (T), we choose a number from {-1, -2, ..., -6} uniformly. Let's call this random variable Z.<BR>\n" );
document.write( "    * X is the random variable representing the final outcome.\r<BR>\n" );
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document.write( "**2. Calculate the Expected Values**\r<BR>\n" );
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document.write( "* **E[Y] (Expected value if heads):**<BR>\n" );
document.write( "    * E[Y] = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) / 10 = 55 / 10 = 5.5<BR>\n" );
document.write( "* **E[Z] (Expected value if tails):**<BR>\n" );
document.write( "    * E[Z] = (-1 - 2 - 3 - 4 - 5 - 6) / 6 = -21 / 6 = -3.5<BR>\n" );
document.write( "* **E[X] (Overall expected value):**<BR>\n" );
document.write( "    * E[X] = E[X|H]P(H) + E[X|T]P(T) = E[Y]P(H) + E[Z]P(T)<BR>\n" );
document.write( "    * E[X] = (5.5)(1/3) + (-3.5)(2/3) = 5.5/3 - 7/3 = -1.5/3 = -0.5\r<BR>\n" );
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document.write( "**3. Calculate the Second Moments**\r<BR>\n" );
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document.write( "* **E[Y²] (Second moment if heads):**<BR>\n" );
document.write( "    * E[Y²] = (1² + 2² + ... + 10²) / 10 = (385) / 10 = 38.5<BR>\n" );
document.write( "* **E[Z²] (Second moment if tails):**<BR>\n" );
document.write( "    * E[Z²] = ((-1)² + (-2)² + ... + (-6)²) / 6 = (91) / 6 = 15.1667 (approximately)<BR>\n" );
document.write( "* **E[X²] (Overall second moment):**<BR>\n" );
document.write( "    * E[X²] = E[Y²]P(H) + E[Z²]P(T)<BR>\n" );
document.write( "    * E[X²] = (38.5)(1/3) + (91/6)(2/3) = 38.5/3 + 91/9 = 12.8333 + 10.1111 = 22.9444 (approximately)\r<BR>\n" );
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document.write( "**4. Calculate the Variance**\r<BR>\n" );
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document.write( "* **Var(X) = E[X²] - (E[X])²**<BR>\n" );
document.write( "* Var(X) = 22.9444 - (-0.5)² = 22.9444 - 0.25 = 22.6944\r<BR>\n" );
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document.write( "**5. Round to Two Decimal Places**\r<BR>\n" );
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document.write( "* Var(X) ≈ 22.69\r<BR>\n" );
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document.write( "Therefore, the variance of X is approximately 22.69.<BR>\n" );
document.write( " </SPAN>\n" );
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