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Thread: Exponential Random Variables

  1. #1
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    Exponential Random Variables

    Let X be an exponential random variable with mean 1. Find the probability density function of $\displaystyle Y = -ln (X)$.

    Can someone walk me through how to do this?

    Exponential random variable has a density function $\displaystyle f(x) = \lambda e^{-\lambda x}$. In this case $\displaystyle \lambda = 1$. How do I find that density function of Y?
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    Quote Originally Posted by Zennie View Post
    Let X be an exponential random variable with mean 1. Find the probability density function of $\displaystyle Y = -ln (X)$.

    Can someone walk me through how to do this?

    Exponential random variable has a density function $\displaystyle f(x) = \lambda e^{-\lambda x}$. In this case $\displaystyle \lambda = 1$. How do I find that density function of Y?
    What techniques have you been taught for finding the pdf of a function of a random variable? Do you have any examples to follow? What have you tried and where do you get stuck?
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    Quote Originally Posted by Zennie View Post
    Let X be an exponential random variable with mean 1. Find the probability density function of $\displaystyle Y = -ln (X)$.

    Can someone walk me through how to do this?

    Exponential random variable has a density function $\displaystyle f(x) = \lambda e^{-\lambda x}$. In this case $\displaystyle \lambda = 1$. How do I find that density function of Y?
    Hint: the cdf of Y is $\displaystyle F_{Y}(y)= P(Y\leq y)$

    now use $\displaystyle Y=-\ln(X)$ above to find the cdf in terms of x. Then differentiate to find the pdf!
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  4. #4
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    Quote Originally Posted by Zennie View Post
    Let X be an exponential random variable with mean 1. Find the probability density function of $\displaystyle Y = -ln (X)$.

    Can someone walk me through how to do this?

    Exponential random variable has a density function $\displaystyle f(x) = \lambda e^{-\lambda x}$. In this case $\displaystyle \lambda = 1$. How do I find that density function of Y?
    Assume that$\displaystyle R=f(x)$ in this case$\displaystyle R = \lambda e^{-\lambda x}$
    we need to proof that$\displaystyle x=F(R).$
    let's do it:
    $\displaystyle
    1-R= \lambda e^{-\lambda x}$
    $\displaystyle Ln(1-R)=Ln( \lambda e^{-\lambda x})$
    $\displaystyle Ln(1-R)=-\lambda x Ln( e) ..........Ln(e)=1$
    x= (-1/ lambda) x Ln(1-R)......... Ln(1-R) same as Ln(R) CZ R<1 , R IS UNIFROM DISTRIBUTION OVER [0,1]
    x= (-1/ lambda) x Ln(R)
    .... THIS IS INVERSE TRANSFORM THECHNIQUE
    HOPE THAT HELPS
    ABDEL
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