# Using Change of Variables to get the distribution function

I have a random variable, call it $X$. This is a uniform random variable that is defined on the $[0,1]$ interval. Now consider its transformation $Z$ defined as: $$Z=klog\times(X)$$ where $k$ is a negative real number. I now wish to calculate the distribution function of $Z$. This is what I have done. Now, by definition, $$F_{Z}(Z)\equiv Pr(z\leq Z)=Pr(klog(X)\leq Z)$$ $$=Pr(logX\geq Z/k)=Pr(X\geq 10^Z/k)=1-Pr(X\leq 10^Z/k)$$ $$=1-(10^Z)/k$$

Is this correct?

## 1 Answer

No. You have a few errors:

• Usually we use the capitol $Z$ to represent the random variable, so $F_Z(Z) = Pr(Z \le z)$.

• Your end result is a distribution function, you need to differentiate this to obtain the density

• The inverse of $\log$ is $\exp$. In statistics, $\log$ is always $\log_e$ not $\log_{10}$.

• Thanks..I agree with the first two points. This variable was generated in a programming language, so I am not sure if we are talking about the natural log (depends what the term "log" corresponds to in the specific language) – ChinG Feb 23 '16 at 16:32
• @ChinG well since you're posting on a statistics site, you should write base 10. It's just an exponential variable either way, but your mean is not necessarily $1/k$. – AdamO Feb 23 '16 at 16:36