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Thread: Distribution functions of Discrete Random Variables

  1. #1
    Senior Member I-Think's Avatar
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    Distribution functions of Discrete Random Variables

    Not sure of my understanding, so asking for help
    2 part question
    If $\displaystyle X$ has a distribution function $\displaystyle F$
    a) What is the distribution function of $\displaystyle e^X$
    b) What is the distribution function of $\displaystyle \alpha{X}+\beta$, where $\displaystyle \alpha$ and $\displaystyle \beta$ are constants, $\displaystyle \alpha \neq{0}$

    Solution attempts
    I know that
    $\displaystyle F(x)=P[X\leq{x}]$

    and $\displaystyle F(a)=\sum_{all x\leq{a}} p(x)$

    Therefore, should I say that $\displaystyle F(x)=P[X\leq{e^x}]$ (ditto for part b) and continue from there? But I'm not sure how to continue, so help please
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  2. #2
    Grand Panjandrum
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    Re: Distribution functions of Discrete Random Variables

    Quote Originally Posted by I-Think View Post
    Not sure of my understanding, so asking for help
    2 part question
    If $\displaystyle X$ has a distribution function $\displaystyle F$
    a) What is the distribution function of $\displaystyle e^X$
    b) What is the distribution function of $\displaystyle \alpha{X}+\beta$, where $\displaystyle \alpha$ and $\displaystyle \beta$ are constants, $\displaystyle \alpha \neq{0}$

    Solution attempts
    I know that
    $\displaystyle F(x)=P[X\leq{x}]$

    and $\displaystyle F(a)=\sum_{all x\leq{a}} p(x)$

    Therefore, should I say that $\displaystyle F(x)=P[X\leq{e^x}]$ (ditto for part b) and continue from there? But I'm not sure how to continue, so help please
    Put $\displaystyle Y=e^X$

    $\displaystyle F_Y(y)=P(Y\le y)=P(\ln(Y)\le \ln(y))=P(X\le \ln(y))=F_X(\ln(y))$

    CB
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  3. #3
    Senior Member I-Think's Avatar
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    Re: Distribution functions of Discrete Random Variables

    Thanks
    So part b should be

    Let $\displaystyle Y=\alpha{X}+\beta$

    So

    $\displaystyle F_Y(y)=P(Y\leq{y})=P(\frac{Y-\beta}{\alpha}\leq{\frac{y-\beta}{\alpha}}=P(X\leq{\frac{y-\beta}{\alpha}})=F_X(\frac{y-\beta}{\alpha})$
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  4. #4
    Grand Panjandrum
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    Re: Distribution functions of Discrete Random Variables

    Yes.

    CB
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