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Expected Monthly Profit

  1. Jun 16, 2017 #1
    1. The problem statement, all variables and given/known data
    A small manufacturing firm sells 1 machine per month with 0.3 probability; it sells 2 machines per month with 0.1 probability; it never sells more than 2 machines per month. If X represents the number of machines sold per month and the monthly profit is 2X2 + 3X + 1 (in thousands of dollars), find the expected monthly profit.

    2. Relevant equations


    3. The attempt at a solution
    E(2X2 + 3X + 1) = ∑(2X2 + 3X + 1)f(x), x = 1, 2
    = (2+3+1)(.3) + (8+6+1)(.1)
    = 1.8 + 1.5
    = 3.3
    3.3 * 1000 = $3,300

    The answer in the back is $3,800.

    If I take my 3.3 and add the mean of E(X) = .3(1) + 2(.1) = 0.5, I get 3.8 * 1000 = $3,800. I'm not sure if this is coincidence or actually how you solve the problem. If it's how you solve the problem, I don't understand why you add the mean of X. If it's not the way you solve it, I'm not sure what to do and would like some hints (but not a solution -- if possible) as to where to go.

    Thanks.
     
  2. jcsd
  3. Jun 16, 2017 #2

    Orodruin

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    You are missing one possible outcome ...

    Edit: Not that that actually fixes the problem ...
     
  4. Jun 16, 2017 #3

    StoneTemplePython

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    So profit ##= 2X^2 + 3X + 1##

    ##E[profit] = E[2X^2 + 3X + 1] = E[2X^2] + E[3X] + E[1]##

    by linearity of expectations.

    This last line should be simplified a bit -- how would you do it? You correctly calculated ##E[X]##. What is ##E[X^2]##? Or if you prefer, what is the Variance -- you can recover ##E[X^2]## from that.
     
  5. Jun 16, 2017 #4

    StoneTemplePython

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    I'm getting an answer a touch higher than the 3.8 (thousand) mentioned as the official one, though.
     
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