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Math Help - p.g.f / density function

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    p.g.f / density function

    (a) A discrete random variable takes on the values 0,1,2 and 3 with probabilities 1/8, 1/4, 1/2 respectively. Determine the p.g.f of the sum of two such random variables which are independent, stating any properties of the p.g.f's that you need to use.

    (b) A random variable X has density given by

    f(x) = 3x^2 0 < x < 1
    0 otherwise

    Let Y be a new random variable defined by Y = X^2. Determine the density function of Y.
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  2. #2
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    Quote Originally Posted by shtaM View Post
    (a) A discrete random variable takes on the values 0,1,2 and 3 with probabilities 1/8, 1/4, 1/2 respectively. Determine the p.g.f of the sum of two such random variables which are independent, stating any properties of the p.g.f's that you need to use.

    (b) A random variable X has density given by

    f(x) = 3x^2 0 < x < 1
    0 otherwise

    Let Y be a new random variable defined by Y = X^2. Determine the density function of Y.
    (a) Apply the following:

    Let Y = X_1 + X_2 where X_1 and X_2 are independent.

    Then G_Y(z) = G_{X_1}(z) \cdot G_{X_2}(z) where G_{X_1}(z) = G_{X_2}(z) = E(z^X) = \sum_{x=0}^3 \Pr(X = 0) \cdot z^x.


    (b) The cdf of Y is given by

    G(y) = \Pr(Y < y) = \Pr(X^2 < y) =\Pr(-\sqrt{y} < X < + \sqrt{y}) = \int_{0}^{\sqrt{y}} 3x^2 \, dx.

    The pdf of Y is given by g(y) = \frac{dG}{dy}.
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