Solved – Is the multiple of a lognormally distributed variable also lognormally distributed

Question

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:

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

Answer 1

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