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Math Help - Expected Value of a Function of a Random Variable Questions

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    Expected Value of a Function of a Random Variable Questions

    1) A tool and die company makes castings for steel stress-monitoring gauges. Their annual profit, Q, in hundreds of thousands of dollars, can be expressed as a function of product demand, y:
    Q(y) = 2(1-e^(-2y))
    Suppose that the demand (in thousands) for their castings follows an exponential pdf, fy(y) = 6e^(-6y), y > 0. Find the company's expected profit.

    2) The hypotenuse, Y, of an isosceles right triangle is a random variable having a uniform pdf over the interval [6,10]. Calculate the expected value of the triangle's area. Do not leave the answer as a function of a. (It shows the triangle with both smaller sides as a and the hypotenuse as Y.)
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    Quote Originally Posted by Janu42 View Post
    1) A tool and die company makes castings for steel stress-monitoring gauges. Their annual profit, Q, in hundreds of thousands of dollars, can be expressed as a function of product demand, y:
    Q(y) = 2(1-e^(-2y))
    Suppose that the demand (in thousands) for their castings follows an exponential pdf, fy(y) = 6e^(-6y), y > 0. Find the company's expected profit.

    [snip]
    E(Q) = 2 E\left(1 - e^{-2y} \right) = 2 \int_0^{+\infty} (1 - e^{-2y}) 6 e^{-6y} \, dy = ....

    where your job is to simplify the integrand and then integrate.
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    Quote Originally Posted by Janu42 View Post
    [snip]
    2) The hypotenuse, Y, of an isosceles right triangle is a random variable having a uniform pdf over the interval [6,10]. Calculate the expected value of the triangle's area. Do not leave the answer as a function of a. (It shows the triangle with both smaller sides as a and the hypotenuse as Y.)
    I assume it's a right isosceles triangle since you talk about a hypotenuse.

    A = \frac{a^2}{2} = \frac{Y^2}{4} using Pythagoras' Theorem.

    E(A) = \frac{1}{4} E(Y^2) = \frac{1}{4} \int_{6}^{10} y^2 \cdot \frac{1}{4} \, dy = ....

    where your job is to simplify and then integrate.
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