Distribution functions of Discrete Random Variables

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November 12th 2011, 08:41 PM
I-Think

Distribution functions of Discrete Random Variables

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

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

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

Therefore, should I say that $F(x)=P[X\leq{e^x}]$ (ditto for part b) and continue from there? But I'm not sure how to continue, so help please
November 12th 2011, 10:20 PM
CaptainBlack

Re: Distribution functions of Discrete Random Variables

Quote:

Originally Posted by I-Think

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

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

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

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

Put $Y=e^X$

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

CB
November 13th 2011, 07:22 AM
I-Think

Re: Distribution functions of Discrete Random Variables

Thanks
So part b should be

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

So

$F_Y(y)=P(Y\leq{y})=P(\frac{Y-\beta}{\alpha}\leq{\frac{y-\beta}{\alpha}}=P(X\leq{\frac{y-\beta}{\alpha}})=F_X(\frac{y-\beta}{\alpha})$
November 13th 2011, 09:53 AM
CaptainBlack

Re: Distribution functions of Discrete Random Variables

Yes.

CB