Distribution functions of Discrete Random Variables

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

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

**CaptainBlack** · November 12th 2011, 10:20 PM

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

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

Thanks
So part b should be

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

So

$F_Y(y)=P(Y\leq{y})=P(\frac{Y-\beta}{\alpha}\leq{\frac{y-\beta}{\alpha}}=P(X\leq{\frac{y-\beta}{\alpha}})=F_X(\frac{y-\beta}{\alpha})$

**CaptainBlack** · November 13th 2011, 09:53 AM

Yes.

CB

Thread: Distribution functions of Discrete Random Variables

LinkBack

Thread Tools

Display

Distribution functions of Discrete Random Variables

Re: Distribution functions of Discrete Random Variables

Re: Distribution functions of Discrete Random Variables

Re: Distribution functions of Discrete Random Variables

Similar Math Help Forum Discussions

help with discrete random variables

Joint Distribution of discrete random variables homework problem

What's the difference between discrete random variables and continous random variable

Discrete Random Variables

Discrete Random Variables

Search Tags