# What is the distribution of an affine transformation of a log-normal variable?

Let $X$ be a log-normal variate and $Y = aX + b$ is the affine transformation X.

Is $Y$ log-normal? I suspect it is not.

Since $X$ is log-normal, its expected value is

$$E[X] = \exp(M + S^2/2)$$

The expected value of $Y$ is:

$$E[Y] = aE[X] + b = a \exp(M + S^2/2) + b$$

To characterize $Y$ as a log-normal variate I should write its mean in the same form of $X$ mean, and obviously reproduce the same approach to others moments.

Does that make sense? or Is there another way to characterize $Y$ as log-normal? For example, using characteristic function.

• By definition $Y$ cannot be lognormal when $b$ is nonzero, for when $b\gt 0$ there is zero probability in the interval $[0,b]$ and when $b\lt 0$ there is positive probability in the interval $[b,0]$, neither of which is true for any lognormal distribution. If you allow nonzero $b$, then this is called a three-parameter lognormal distribution. – whuber Aug 18 '16 at 14:52
• Also worth mentioning is that $Y=aX$ would be log-normal for a scalar $a$, (Easy to see by taking log of both sides of the equation). More generally, multiplying for log-normal distributed random variables is basically equivalent to adding normal distributed random variables. – Matthew Gunn Aug 18 '16 at 15:12
• The particular three parameter lognormal whuber refers to is also sometimes called the shifted lognormal. – Glen_b Aug 19 '16 at 2:37