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Function of random variable, limits of integration

  1. Apr 28, 2013 #1
    1. The problem statement, all variables and given/known data

    X is uniformly distributed over [-1,1]. Compute the density function f(y) of Y = 2X2 + 1.


    2. Relevant equations



    3. The attempt at a solution

    FY(Y) = P(Y < y) = P(2X2 + 1 < y) = P(X < +[itex]\sqrt{1/2(y-1)}[/itex] = FX(+[itex]\sqrt{1/2(y-1)}[/itex])

    We have that f(x) = 0.5 for -1 < x < 1, so we should integrate f(x) from -1 to +[itex]\sqrt{1/2(y-1)}[/itex] to get FY(y), and then take the derivative of that to get f(y).

    But how do I deal with the + in front of the square root? What does P(X < +[itex]\sqrt{1/2(y-1)}[/itex] even mean? "The probability that X is smaller than plus minus x" doesn't seem to make much sense to me!!
     
  2. jcsd
  3. Apr 28, 2013 #2
    I guess it means X is BETWEEN -[itex]\sqrt{1/2(y-1)}[/itex] and +[itex]\sqrt{1/2(y-1)}[/itex]? But how do I figure out which bounds y has in that case?
     
  4. Apr 28, 2013 #3

    HallsofIvy

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    You are told that X lies between -1 and 1. [itex]2X^2+ 1[/itex] is a parabola which has a minimum at Y= 1 (when X= 0) and a maximum of Y= 3 (at x= -1 and 1).

    Now, the problem asks you to find the "probability density function" for Y so why are you integrating at all?
     
  5. Apr 28, 2013 #4
    I'm trying to find F(y) so I can take the derivative and get f(y). Basically trying to follow lecture notes...
     
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