document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182767>Question 1182767</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Problem 2 (13%) <BR>\n" );
document.write( "Consider the random variable X and its probability mass function: <BR>\n" );
document.write( "𝑋: 70 80 90 <BR>\n" );
document.write( "𝑃(𝑋 = 𝑥): 0.3 0.45 0.25 <BR>\n" );
document.write( "a- Find the expected value of this random variable. <BR>\n" );
document.write( "b- Find the cumulative distribution of this random. <BR>\n" );
document.write( "c- Calculate P(𝑋 ≥ 80) in two methods: Use the cumulative distribution and the probability mass function. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #813257 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=813257\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "Given table<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >X</TD><TD >70</TD><TD >80</TD><TD >90</TD></tr><tr><TD >P(X)</TD><TD >0.3</TD><TD >0.45</TD><TD >0.25</TD></tr>\n" );
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document.write( "Form a new row that consists of multiplying the X and P(X) values<BR>\n" );
document.write( "Example: 70*0.3 = 21 in the first column<BR>\n" );
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document.write( "<tr><TD >X</TD><TD >70</TD><TD >80</TD><TD >90</TD></tr><tr><TD >P(X)</TD><TD >0.3</TD><TD >0.45</TD><TD >0.25</TD></tr><tr><TD >X*P(X)</TD><TD >21</TD><TD >36</TD><TD >22.5</TD></tr>\n" );
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document.write( "Add up everything in that third row: 21+36+22.5 = 79.5<BR>\n" );
document.write( "This is the expected value.\r<BR>\n" );
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document.write( "Answer: 79.5\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Given table<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >X</TD><TD >70</TD><TD >80</TD><TD >90</TD></tr><tr><TD >P(X)</TD><TD >0.3</TD><TD >0.45</TD><TD >0.25</TD></tr>\n" );
document.write( "</TABLE>Let's define a function C(X) such that it computes the cumulative probability up to and including that X value.\r<BR>\n" );
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document.write( "We'll define C(X) like so<BR>\n" );
document.write( "C(70) = P(70) = 0.3<BR>\n" );
document.write( "C(80) = P(70)+P(80) = 0.3+0.45 = 0.75<BR>\n" );
document.write( "C(90) = P(70)+P(80)+P(90) = 0.3+0.45+0.25 = 1<BR>\n" );
document.write( "So again, we define <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28X%29+=+P%28X+%3C=+k%29 ALIGN=MIDDLE  ALT=\"C%28X%29+=+P%28X+%3C=+k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where k is a value in the domain of X.\r<BR>\n" );
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document.write( "So we have<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >X</TD><TD >70</TD><TD >80</TD><TD >90</TD></tr><tr><TD >P(X)</TD><TD >0.3</TD><TD >0.45</TD><TD >0.25</TD></tr><tr><TD >C(X)</TD><TD >0.3</TD><TD >0.75</TD><TD >1</TD></tr>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "Given table<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >X</TD><TD >70</TD><TD >80</TD><TD >90</TD></tr><tr><TD >P(X)</TD><TD >0.3</TD><TD >0.45</TD><TD >0.25</TD></tr>\n" );
document.write( "</TABLE>We see that the probability of getting 80 or higher, based on that table above, is 0.45+0.25 = 0.70<BR>\n" );
document.write( "Simply add the probability values for X = 80 or larger.<BR>\n" );
document.write( "There's a 70% chance of this happening.\r<BR>\n" );
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document.write( "Or, we could note that C(70) = 0.30<BR>\n" );
document.write( "From this, we could then say <BR>\n" );
document.write( "P(𝑋 ≥ 80) = 1 - C(70)<BR>\n" );
document.write( "P(𝑋 ≥ 80) = 1 - 0.30<BR>\n" );
document.write( "P(𝑋 ≥ 80) = 0.70\r<BR>\n" );
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document.write( "This works because<BR>\n" );
document.write( "P(70) + P(80) + P(90) = 1<BR>\n" );
document.write( "The \"P(80) + P(90)\" portion is what we want while C(70) represents P(70)\r<BR>\n" );
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document.write( "In other words,<BR>\n" );
document.write( "P(70) + P(80) + P(90) = 1<BR>\n" );
document.write( "P(70) + [ P(80) + P(90) ] = 1<BR>\n" );
document.write( "C(70) + P(𝑋 ≥ 80) = 1<BR>\n" );
document.write( "P(𝑋 ≥ 80) = 1 - C(70)<BR>\n" );
document.write( "P(𝑋 ≥ 80) = 1 - 0.3<BR>\n" );
document.write( "P(𝑋 ≥ 80) = 0.7\r<BR>\n" );
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document.write( "Answer: 0.70<BR>\n" );
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