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Thread: Transformations of random variables with the exponential distribution

  1. #1
    Senior Member chella182's Avatar
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    Transformations of random variables with the exponential distribution

    Just posted a similar thread, but I'm not sure if you're allowed multiple questions in one thread (sorry if you are!)

    $\displaystyle X$ is exponentially distributed with parameter $\displaystyle \lambda$. For the random variable $\displaystyle Y=\frac{1}{X}$
    i) State the range of values $\displaystyle Y$ can take.
    ii) Obtain the distribution function of $\displaystyle Y$, $\displaystyle F_{Y}(y)$.
    iii) Obtain the probability density function of $\displaystyle Y$, $\displaystyle f_{Y}(y)$.
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  2. #2
    MHF Contributor matheagle's Avatar
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    First of all I would like to know how you write your exponential distribution.
    And the range is $\displaystyle (0,\infty)$

    $\displaystyle F_Y(y)=P(Y\le y)=P(1/X\le y)=P(X\ge 1/y)=\lambda\int_{1/y}^{\infty}e^{-x\lambda} dx$
    Last edited by matheagle; Nov 26th 2009 at 02:37 PM.
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  3. #3
    Senior Member chella182's Avatar
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    $\displaystyle \lambda e^{-\lambda x}$ is how we use the exponential distribution.
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