6
$\begingroup$

If $X$ is distributed normally $N(\mu,\sigma^2)$ then the variable $Y = \exp(X)$ is lognormally distributed.

If the variable $Y$ is multiplied by some constant $C$:

$$D = CY$$

Is the variable $D$ also lognormally distributed?

Yes it is since:

$$D = C\exp(X)$$ $$\log(D) = \log(C\exp(X)) = \ln(C) + X$$ $$X + \ln(C) \sim N(\mu + ln(C), \sigma^2)$$ $$D \sim \ln N(\mu + ln(C), \sigma^2)$$

$\endgroup$
3
  • 2
    $\begingroup$ If $Z$ is normally distributed, what's the distribution of $Z+a$? $\endgroup$
    – Glen_b
    Commented Jul 19, 2016 at 10:51
  • $\begingroup$ $Z$ will be normal $N(\mu_Z+\alpha,\sigma_Z^2)$. So I suppose this means that $D$ is my question is lognormally distributed. $\endgroup$
    – egg
    Commented Jul 19, 2016 at 10:53
  • $\begingroup$ Yes, Z+a is normal so exp(Z+a) is lognormal. If you want to answer yourself, go ahead. If not I'll probably put an answer sometime soon $\endgroup$
    – Glen_b
    Commented Jul 19, 2016 at 22:38

1 Answer 1

4
$\begingroup$

The multiple of a lognormal variable is also lognormally distributed.

$$X \sim \mathcal{N}(\mu,\sigma^2)$$ $$Y = e^X$$ $$CY = Ce^X$$

The random variable $CY$ is lognormally distributed since:

$$\ln(CY) = \ln(Ce^X)=\ln(C)+\ln(e^X)=\ln(C)+X$$

Note that:

$$\ln(C) + X \sim \mathcal{N}(\mu + \ln(C), \sigma^2)$$

Therefore $CY$ is lognormally distributed $CY \sim \ln \mathcal{N}(\mu + \ln(C), \sigma^2)$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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