document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191307>Question 1191307</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Company ABC estimates the net profit on a new product it is launching to be Rs. 3,000,000 if it is successful ; Rs. 1,000,000 if it is 'moderately successful' and a loss of Rs. 1,000,000 if it is 'unsuccessful'. The firm assigns the following probabilities to the different possibilities :Successful 0-15, 'moderately successful' 0-25 and unsuccessful 0-60. Find the expected value and variance of the net profit. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849154 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=849154\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the expected value and variance of the net profit:\r<BR>\n" );
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document.write( "**1. Define the Random Variable:**\r<BR>\n" );
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document.write( "Let X be the random variable representing the net profit.\r<BR>\n" );
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document.write( "**2. Create the Probability Distribution:**\r<BR>\n" );
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document.write( "| Outcome | Net Profit (X) | Probability (P(X)) |<BR>\n" );
document.write( "|---|---|---|<BR>\n" );
document.write( "| Successful | Rs. 3,000,000 | 0.15 |<BR>\n" );
document.write( "| Moderately Successful | Rs. 1,000,000 | 0.25 |<BR>\n" );
document.write( "| Unsuccessful | -Rs. 1,000,000 | 0.60 |\r<BR>\n" );
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document.write( "**3. Calculate the Expected Value (Mean):**\r<BR>\n" );
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document.write( "The expected value E(X) is calculated as the sum of each possible value of X multiplied by its probability:\r<BR>\n" );
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document.write( "E(X) = (3,000,000 * 0.15) + (1,000,000 * 0.25) + (-1,000,000 * 0.60)<BR>\n" );
document.write( "E(X) = 450,000 + 250,000 - 600,000<BR>\n" );
document.write( "E(X) = 100,000\r<BR>\n" );
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document.write( "Therefore, the expected net profit is Rs. 100,000.\r<BR>\n" );
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document.write( "**4. Calculate the Variance:**\r<BR>\n" );
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document.write( "The variance Var(X) is calculated as the expected value of the squared difference between each value of X and the mean:\r<BR>\n" );
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document.write( "Var(X) = Σ[(X - E(X))² * P(X)]\r<BR>\n" );
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document.write( "Var(X) = (3,000,000 - 100,000)² * 0.15 + (1,000,000 - 100,000)² * 0.25 + (-1,000,000 - 100,000)² * 0.60<BR>\n" );
document.write( "Var(X) = (2,900,000)² * 0.15 + (900,000)² * 0.25 + (-1,100,000)² * 0.60<BR>\n" );
document.write( "Var(X) = 8,410,000,000,000 * 0.15 + 810,000,000,000 * 0.25 + 1,210,000,000,000 * 0.60<BR>\n" );
document.write( "Var(X) = 1,261,500,000,000 + 202,500,000,000 + 726,000,000,000<BR>\n" );
document.write( "Var(X) = 2,190,000,000,000\r<BR>\n" );
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document.write( "Therefore, the variance of the net profit is Rs. 2,190,000,000,000.\r<BR>\n" );
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document.write( "**5. Calculate the Standard Deviation:**\r<BR>\n" );
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document.write( "The standard deviation SD(X) is the square root of the variance:\r<BR>\n" );
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document.write( "SD(X) = √Var(X)<BR>\n" );
document.write( "SD(X) = √2,190,000,000,000<BR>\n" );
document.write( "SD(X) ≈ Rs. 1,479,864.86\r<BR>\n" );
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document.write( "**Summary of Results:**\r<BR>\n" );
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document.write( "* **Expected Value (Mean):** Rs. 100,000<BR>\n" );
document.write( "* **Variance:** Rs. 2,190,000,000,000<BR>\n" );
document.write( "* **Standard Deviation:** Rs. 1,479,864.86<BR>\n" );
document.write( " </SPAN>\n" );
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