Q.4.2 - A First Course in Probability

Question

If X has distribution function F, what is the distribution function of  $e^X$?

Step-by-Step Solution

Answer

 In the given information The distribution of $e^X$ can be written as composition of logarithm function and  distribution function of X

Step 1:Given Information

Take any $a\in supp{e}^X$(the support of $e^X$ is not specified since we are not given  the support of X ).We have that

$F_{e^X}\left(a\right)=P\left(e^X\le a\right)=P(X\le \log (a\left)\right)=F_X\left(\log a\right)$

Step 2: Explanation

 $e^X$ can be written as composition of logarithm function and  distribution function of X

Step 3: Final Answer

The distribution of $e^X$ can be written as composition of logarithm function and distribution function of X