document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177122>Question 1177122</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The yearly inflation rate in the country follows the probability density function:<BR>\n" );
document.write( "f(x)   =  c/𝑥^3 for 𝑥 > 3 <BR>\n" );
document.write( "f(x)   =  0     𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒<BR>\n" );
document.write( "Where c is a constant. <BR>\n" );
document.write( "a) Find the value of c. <BR>\n" );
document.write( "b) Find an approximate probability that exactly 1 out of 5 coming years have yearly inflation rate greater than 20. (Hint: p is small)\r<BR>\n" );
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document.write( "c) Find 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 #850518 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=850518\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll find the value of c, the probability of a specific inflation rate outcome, and the cumulative distribution function (CDF).<BR>\n" );
document.write( "```python?code_reference&code_event_index=3<BR>\n" );
document.write( "from sympy import integrate, Symbol, oo\r<BR>\n" );
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document.write( "# Define the variable<BR>\n" );
document.write( "x = Symbol('x')\r<BR>\n" );
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document.write( "# a) Find the value of c<BR>\n" );
document.write( "c = 1 / integrate(x**-3, (x, 3, oo))<BR>\n" );
document.write( "print(f\"a) The value of c is: {c}\")\r<BR>\n" );
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document.write( "# b) Find the probability that the inflation rate is greater than 20<BR>\n" );
document.write( "p = integrate(c * x**-3, (x, 20, oo))<BR>\n" );
document.write( "print(f\"b) The probability that the inflation rate is greater than 20 is: {p}\")\r<BR>\n" );
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document.write( "# Calculate the probability that exactly one out of five years has an inflation rate greater than 20<BR>\n" );
document.write( "n = 5<BR>\n" );
document.write( "k = 1<BR>\n" );
document.write( "prob_1_out_of_5 = (p ** k) * ((1 - p) ** (n - k)) * (5 / 1)  # binomial probability formula<BR>\n" );
document.write( "print(f\"The probability that exactly one out of five years has an inflation rate greater than 20 is: {prob_1_out_of_5}\")\r<BR>\n" );
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document.write( "# c) Find the cumulative distribution function (CDF)<BR>\n" );
document.write( "cdf = integrate(c * x**-3, (x, 3, x))<BR>\n" );
document.write( "print(f\"c) The cumulative distribution function (CDF) is: {cdf}\")<BR>\n" );
document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=3<BR>\n" );
document.write( "a) The value of c is: 18<BR>\n" );
document.write( "b) The probability that the inflation rate is greater than 20 is: 9/400<BR>\n" );
document.write( "The probability that exactly one out of five years has an inflation rate greater than 20 is: 0.102711621801270<BR>\n" );
document.write( "c) The cumulative distribution function (CDF) is: 1 - 9/x**2\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "Here are the results:\r<BR>\n" );
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document.write( "* **a) Value of c:**<BR>\n" );
document.write( "   The constant c in the probability density function is 18.\r<BR>\n" );
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document.write( "* **b) Probability of exactly one year out of five with inflation greater than 20:**<BR>\n" );
document.write( "   The probability that exactly one out of the next five years will have an inflation rate greater than 20 is approximately 0.1027.\r<BR>\n" );
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document.write( "* **c) Cumulative Distribution Function (CDF):**<BR>\n" );
document.write( "   The cumulative distribution function (CDF) for the inflation rate (x) is:\r<BR>\n" );
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document.write( "   CDF(x) = 0, for x ≤ 3<BR>\n" );
document.write( "   CDF(x) = 1 - 9/x^2, for x > 3\r<BR>\n" );
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document.write( "Let me know if you have any further questions or need more assistance with this or another problem! </SPAN>\n" );
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