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

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 $\displaystyle X$ 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

Re: Distribution functions of Discrete Random Variables

Thanks

So part b should be

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

So

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

Re: Distribution functions of Discrete Random Variables