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Math Help - continuous random variable help?

  1. #1
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    Smile continuous random variable help?

    The monthly profit of Company I can be modeled by a continuous random variable with density function f . Company II has a monthly …profit that is twice that of Company I. Determine the probability density function of the monthly profit of Company II.

    a) (1/2) * f(x / 2)
    b) f(x / 2)
    c) 2 * f(x / 2)
    d) 2 * f(x)
    e) 2 * f(2x)

    so far for a, I have

    Let Company-I profit be Y and that of Company-II be X. Then X = 2Y, or Y=X/2, dY=dX/2 so that

    f(y)dy = f(x/2)dx/2.

    but I'm not sure if that makes sense? can anyone shed some light? thanks in advance.
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  2. #2
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    Quote Originally Posted by Intsecxtanx View Post
    The monthly profit of Company I can be modeled by a continuous random variable with density function f . Company II has a monthly …profit that is twice that of Company I. Determine the probability density function of the monthly profit of Company II.

    a) (1/2) * f(x / 2)
    b) f(x / 2)
    c) 2 * f(x / 2)
    d) 2 * f(x)
    e) 2 * f(2x)

    so far for a, I have

    Let Company-I profit be Y and that of Company-II be X. Then X = 2Y, or Y=X/2, dY=dX/2 so that

    f(y)dy = f(x/2)dx/2.

    but I'm not sure if that makes sense? can anyone shed some light? thanks in advance.
    x = 2y \Rightarrow y = \frac{x}{2}.

    Therefore f_X = f_Y(y) \cdot \left| \frac{dy}{dx}\right| = \frac{1}{2} f\left( \frac{x}{2}\right).
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