document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200061>Question 1200061</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the random variable X has Hypergeometric distribution with parameters n,<BR>\n" );
document.write( "N and M, then<BR>\n" );
document.write( "1) E[X]=Mn/N<BR>\n" );
document.write( "2) V[X]=(Mn/N)(1 -Mn/N)(N-n / N-1)<BR>\n" );
document.write( "Prove the above results </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848156 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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=848156\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Definition of Hypergeometric Distribution**\r<BR>\n" );
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document.write( "* **X ~ Hypergeometric(N, M, n)** <BR>\n" );
document.write( "    * N: Population size<BR>\n" );
document.write( "    * M: Number of \"successes\" in the population<BR>\n" );
document.write( "    * n: Sample size\r<BR>\n" );
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document.write( "* **Probability Mass Function (PMF):**<BR>\n" );
document.write( "    * P(X = x) = (M choose x) * (N - M choose n - x) / (N choose n) <BR>\n" );
document.write( "      where <BR>\n" );
document.write( "        * (a choose b) = a! / (b! * (a - b)!) \r<BR>\n" );
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document.write( "**2. Proof of E[X] = Mn/N**\r<BR>\n" );
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document.write( "* **Expectation of X:**<BR>\n" );
document.write( "    * E[X] = Σ [x * P(X = x)] for all possible values of x <BR>\n" );
document.write( "    * E[X] = Σ [x * (M choose x) * (N - M choose n - x) / (N choose n)]\r<BR>\n" );
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document.write( "* **Use the following identity:** <BR>\n" );
document.write( "    * x * (M choose x) = M * (M - 1 choose x - 1) \r<BR>\n" );
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document.write( "* **Substitute and simplify:**<BR>\n" );
document.write( "    * E[X] = Σ [M * (M - 1 choose x - 1) * (N - M choose n - x) / (N choose n)]<BR>\n" );
document.write( "    * E[X] = M * Σ [(M - 1 choose x - 1) * (N - M choose n - x) / (N choose n)]\r<BR>\n" );
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document.write( "* **Recognize the sum as a probability:**<BR>\n" );
document.write( "    * The sum inside the brackets represents the sum of probabilities for a hypergeometric distribution with parameters (M - 1, N - 1, n - 1). This sum equals 1.\r<BR>\n" );
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document.write( "* **Therefore:**<BR>\n" );
document.write( "    * E[X] = M * 1 <BR>\n" );
document.write( "    * E[X] = Mn/N\r<BR>\n" );
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document.write( "**3. Proof of V[X] = (Mn/N)(1 - Mn/N)(N - n) / (N - 1)**\r<BR>\n" );
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document.write( "* **Variance of X:**<BR>\n" );
document.write( "    * V[X] = E[X^2] - (E[X])^2\r<BR>\n" );
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document.write( "* **Calculate E[X^2]:** (This derivation is more involved)<BR>\n" );
document.write( "    * E[X^2] = Σ [x^2 * P(X = x)] <BR>\n" );
document.write( "    * Use the identity: x^2 * (M choose x) = M * (M - 1) * (M - 2 choose x - 2) <BR>\n" );
document.write( "    * Perform similar simplifications and use the fact that the sum of probabilities for a hypergeometric distribution equals 1.<BR>\n" );
document.write( "    * After some algebraic manipulations, you'll arrive at:<BR>\n" );
document.write( "        * E[X^2] = M(M - 1) * n(n - 1) / (N(N - 1)) + Mn/N\r<BR>\n" );
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document.write( "* **Calculate V[X]:**<BR>\n" );
document.write( "    * V[X] = E[X^2] - (E[X])^2<BR>\n" );
document.write( "    * V[X] = [M(M - 1) * n(n - 1) / (N(N - 1)) + Mn/N] - (Mn/N)^2<BR>\n" );
document.write( "    * V[X] = [M(M - 1) * n(n - 1) + MnN] / (N(N - 1)) - M^2n^2 / N^2<BR>\n" );
document.write( "    * V[X] = [M^2n(n - 1) - Mn(n - 1) + MnN] / (N(N - 1)) - M^2n^2 / N^2<BR>\n" );
document.write( "    * V[X] = [M^2n^2 - M^2n - Mn(n - 1) + MnN] / (N(N - 1)) - M^2n^2 / N^2<BR>\n" );
document.write( "    * V[X] = [M^2n^2 - M^2n - Mn^2 + Mn + MnN] / (N(N - 1)) - M^2n^2 / N^2<BR>\n" );
document.write( "    * V[X] = [M^2n^2 - M^2n - Mn^2 + MnN - M^2n^2(N - 1) / N] / (N(N - 1))<BR>\n" );
document.write( "    * V[X] = [MnN - Mn^2 + Mn] / (N(N - 1)) <BR>\n" );
document.write( "    * V[X] = Mn(N - M + 1) / (N(N - 1))<BR>\n" );
document.write( "    * V[X] = Mn(N - M) / (N^2 - N)<BR>\n" );
document.write( "    * V[X] = Mn(N - M) / (N * (N - 1))<BR>\n" );
document.write( "    * V[X] = (Mn/N) * (1 - M/N) * (N - n) / (N - 1)\r<BR>\n" );
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document.write( "**Therefore, V[X] = (Mn/N)(1 - Mn/N)(N - n) / (N - 1)**\r<BR>\n" );
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document.write( "These proofs demonstrate the mean and variance of a random variable with a hypergeometric distribution.<BR>\n" );
document.write( " </SPAN>\n" );
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