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?


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 $\log_{10}$.

  • $\begingroup$ 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) $\endgroup$ – ChinG Feb 23 '16 at 16:32
  • 1
    $\begingroup$ @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 $1/k$. $\endgroup$ – AdamO Feb 23 '16 at 16:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.