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Thread: Suman Thapa

  1. #1
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    Suman Thapa

    I need a help to solve the question regarding normal distribution. The question is :
    If X and Y are standard normal distribution and are independent then how to find the pdf and mgf of XY?
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  2. #2
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    Re: Suman Thapa

    The (univariate) normal PDF is given by $\displaystyle f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^-{\tfrac{(x-\mu)^2} {2\sigma^2}} $.

    When $\displaystyle X$ is standard normal, $\displaystyle \mu=0 \text{ and } \sigma=1$, so that the standard normal PDF is $\displaystyle f_X(x) = \frac{1}{\sqrt{2\pi}}e^{-{\tfrac{x^2} {2}}} $.

    $\displaystyle Y$ is also a standard normal random variable, so its PDF is similarly $\displaystyle f_Y(y) = \frac{1}{\sqrt{2\pi}}e^{-{\tfrac{y^2} {2}}} $.

    Now, If $\displaystyle X $ and $\displaystyle Y$ are INDEPENDENT, then $\displaystyle f_X(x) \cdot f_Y(y) = f_{X,Y}(x,y)$.

    So, $\displaystyle f_{X,Y}(x,y) = f_X(x) f_Y(y) = \left(\frac{1}{\sqrt{2\pi}}e^{-{\tfrac{x^2} {2}}}\right)\left(\frac{1}{\sqrt{2\pi}}e^{-{\tfrac{y^2} {2}}}\right) = \frac{1}{2\pi}e^{-{\tfrac{x^2+y^2} {2}}} $.
    Thanks from sumanthapa
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  3. #3
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    Re: Suman Thapa

    Thanks for the solution. But I mean, if Z = XY (product of X and Y), where X and Y are standard normal distribution, then how to find the pdf and moment generating function(mgf) of Z? Could you please help me? Thanks.
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