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.

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    $\begingroup$ 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. $\endgroup$ – whuber Aug 18 '16 at 14:52
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    $\begingroup$ 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. $\endgroup$ – Matthew Gunn Aug 18 '16 at 15:12
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    $\begingroup$ The particular three parameter lognormal whuber refers to is also sometimes called the shifted lognormal. $\endgroup$ – Glen_b Aug 19 '16 at 2:37

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