document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1173951>Question 1173951</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let X be the random variable denoting the number of forms required to be filled out by a <BR>\n" );
document.write( "contractor for participation in contract bids, where the values of X are 1, 2, 3, 4, and 5, and <BR>\n" );
document.write( "suppose that the respective probabilities are proportional to X, that is ,<BR>\n" );
document.write( " . Then calculate a) The value of the constant c <BR>\n" );
document.write( "b) d) Mean and variance of the distribution.<BR>\n" );
document.write( "e) Cumulative distribution function (cdf) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850713 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=850713\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Understanding the Problem**\r<BR>\n" );
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document.write( "* X is a random variable representing the number of forms (1, 2, 3, 4, 5).<BR>\n" );
document.write( "* The probabilities are proportional to X: P(X=x) = cx, where c is a constant.\r<BR>\n" );
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document.write( "**a) Finding the Value of the Constant c**\r<BR>\n" );
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document.write( "* The sum of all probabilities must equal 1.<BR>\n" );
document.write( "* Therefore: P(1) + P(2) + P(3) + P(4) + P(5) = 1<BR>\n" );
document.write( "* Substitute P(x) = cx: c(1) + c(2) + c(3) + c(4) + c(5) = 1<BR>\n" );
document.write( "* Simplify: c(1 + 2 + 3 + 4 + 5) = 1<BR>\n" );
document.write( "* c(15) = 1<BR>\n" );
document.write( "* c = 1/15\r<BR>\n" );
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document.write( "**b) Probability Distribution**\r<BR>\n" );
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document.write( "* Now that we know c = 1/15, we can find the probabilities:<BR>\n" );
document.write( "    * P(X=1) = (1/15) * 1 = 1/15<BR>\n" );
document.write( "    * P(X=2) = (1/15) * 2 = 2/15<BR>\n" );
document.write( "    * P(X=3) = (1/15) * 3 = 3/15 = 1/5<BR>\n" );
document.write( "    * P(X=4) = (1/15) * 4 = 4/15<BR>\n" );
document.write( "    * P(X=5) = (1/15) * 5 = 5/15 = 1/3\r<BR>\n" );
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document.write( "**d) Mean (μ) and Variance (σ^2)**\r<BR>\n" );
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document.write( "* **Mean (μ):**<BR>\n" );
document.write( "    * μ = E(X) = Σ[x * P(X=x)]<BR>\n" );
document.write( "    * μ = (1 * 1/15) + (2 * 2/15) + (3 * 3/15) + (4 * 4/15) + (5 * 5/15)<BR>\n" );
document.write( "    * μ = (1 + 4 + 9 + 16 + 25) / 15<BR>\n" );
document.write( "    * μ = 55 / 15 = 11/3 ≈ 3.6667\r<BR>\n" );
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document.write( "* **Variance (σ^2):**<BR>\n" );
document.write( "    * σ^2 = E(X^2) - μ^2<BR>\n" );
document.write( "    * First, find E(X^2):<BR>\n" );
document.write( "        * E(X^2) = Σ[x^2 * P(X=x)]<BR>\n" );
document.write( "        * E(X^2) = (1^2 * 1/15) + (2^2 * 2/15) + (3^2 * 3/15) + (4^2 * 4/15) + (5^2 * 5/15)<BR>\n" );
document.write( "        * E(X^2) = (1 + 8 + 27 + 64 + 125) / 15<BR>\n" );
document.write( "        * E(X^2) = 225 / 15 = 15<BR>\n" );
document.write( "    * Now, calculate σ^2:<BR>\n" );
document.write( "        * σ^2 = 15 - (11/3)^2<BR>\n" );
document.write( "        * σ^2 = 15 - 121/9<BR>\n" );
document.write( "        * σ^2 = (135 - 121) / 9<BR>\n" );
document.write( "        * σ^2 = 14 / 9 ≈ 1.5556\r<BR>\n" );
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document.write( "**e) Cumulative Distribution Function (CDF)**\r<BR>\n" );
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document.write( "* The CDF, F(x) = P(X ≤ x), is the probability that X is less than or equal to x.\r<BR>\n" );
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document.write( "* F(1) = P(X ≤ 1) = P(X=1) = 1/15<BR>\n" );
document.write( "* F(2) = P(X ≤ 2) = P(X=1) + P(X=2) = 1/15 + 2/15 = 3/15 = 1/5<BR>\n" );
document.write( "* F(3) = P(X ≤ 3) = P(X=1) + P(X=2) + P(X=3) = 3/15 + 3/15 = 6/15 = 2/5<BR>\n" );
document.write( "* F(4) = P(X ≤ 4) = P(X=1) + P(X=2) + P(X=3) + P(X=4) = 6/15 + 4/15 = 10/15 = 2/3<BR>\n" );
document.write( "* F(5) = P(X ≤ 5) = P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) = 10/15 + 5/15 = 15/15 = 1\r<BR>\n" );
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document.write( "* Therefore, the CDF is:\r<BR>\n" );
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document.write( "    * F(x) = { 0, if x < 1<BR>\n" );
document.write( "               { 1/15, if 1 ≤ x < 2<BR>\n" );
document.write( "               { 1/5, if 2 ≤ x < 3<BR>\n" );
document.write( "               { 2/5, if 3 ≤ x < 4<BR>\n" );
document.write( "               { 2/3, if 4 ≤ x < 5<BR>\n" );
document.write( "               { 1, if x ≥ 5<BR>\n" );
document.write( " </SPAN>\n" );
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