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Let $ln(X)\sim N(\mu,\sigma^2)$ and $c$ be a constant such that $X+c$ follows a shifted lognormal distribution with parameters $\mu$, $\sigma^2$, and $c$. What I would like to know is the distribution of $\frac{1}{X+c}$. I know that in the special case where $c=0$, $\frac{1}{X}\sim lognormal(-\mu,\sigma^2)$, but what about the more general case?

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If $Y = 1/(X+c)$, $c \in \mathbb{R}$, and $X \sim \text{Lognormal}(\mu,\sigma^2)$, then $$ f_Y(y) = f_X(1/y-c)|y^{-2}|. $$ If you're looking for the name of the distribution, I can't help you there. You can see it isn't another lognormal random variable, though.

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You can relate this to Johnson's $S_B$-distribution

$$z = \gamma + \delta \log \left(\frac{x-\xi}{\xi+\lambda-x} \right)$$

  • with $\xi = 0$, $\lambda = 1/c$

    from which follows with some rearrancements

    $$\frac{z-\gamma}{\delta}= y = \log \left(\frac{cx}{1-cx} \right) \sim N(-\frac{\gamma}{\delta}, \frac{1}{\delta^2})$$

  • and with $\delta = 1/\sigma$ and $\gamma = \frac{\mu-\ln(c)}{\sigma}$

$$x = \frac{1}{c+e^{-y+\ln(c)}} = \frac{1}{c+e^{y^\prime}}\\ \text{with} \quad y^\prime = \ln(c)-y \sim N\left(\mu=\ln(c)+\frac{\gamma}{\delta},\sigma^2= \frac{1}{\delta^2}\right)$$

You could also see it as a scaled logit-/logistic-normal distribution

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