# Distribution of transformation of normally distributed variable

Apart from lognormal, is there any other convenient distribution that is obtained from transformation of a normally distributed variable and has the support of [0,+infinity)?

For $Z\sim N(\mu,\sigma^2)$, $(Z-\mu)^2$ has a gamma distribution, $(\frac{Z-\mu}{\sigma})^2$ has a $\chi^2_1$ distribution.

$|\frac{Z-\mu}{\sigma}|$ has a chi(1)-distribution.

Of course, it depends on what class of transformations you'd consider. If $\Phi$ is the standard normal cdf, and $F$ is an invertible continuous CDF for a non-negative random variable, then $F^{-1}(\Phi(\frac{Z-\mu}{\sigma}))$ has distribution $F$, so in principle there's a very large number of them indeed.

• Thanks for your response. It is a reasonable answer to the question but not exactly what I wanted. Here is a better wording of the question I had in my mind. Suppose for some transformation function "f", f(Z) is normally distributed where Z is a random variable with support [0,infinity). Apart from log what other forms for "f" leads to a "convenient" distribution for Z? Another way of asking my question: suppose I have some non-negative data. what transformation of the data could give a normally distributed variable while I also have a convenient distribution for the original data. – user41838 Nov 7 '14 at 2:25
• If you would like to modify your question, I could delete my answer (though the last paragraph could stand if I simply invert the transformation $Y=F^{-1}(\Phi(Z))$ to $Z=\Phi^{-1}(F(Y))$). Alternatively, you could post a new question. – Glen_b Nov 7 '14 at 3:22

Basically, you can transform any distribution $F$ with support on ${\mathbb R}$ by using an exponential transformation, leading to a log-$F$ distribution. In this line, you can transform any location-scale distribution such as the Logistic, Exponential power, Student-t, skew elliptical, ... The use of other distributions than normal allows you to capture other features. For example, if you use the log-Student-$t$ distribution, you can model heavier tails. On the other hand, if you use the log-skew-normal, you can model different shapes around the mode, with tails close (but not equal) to the normal ones.

More specifically, you can create the distribution you want by taking any distribution $F$ with density $f$ with support on ${\mathbb R}$, and using the transformation:

\begin{eqnarray} LF(x;\mu,\sigma) &=& F\left[\frac{\log(x)-\mu}{\sigma}\right],\\ lf(x;\mu,\sigma) &=& \frac{1}{\sigma x}f\left[\frac{\log(x)-\mu}{\sigma}\right]. \end{eqnarray}

If $F$ is the normal distribution, then you obtain the log-normal distribution. If $f$ is the Student-$t$, then you obtain the log-Student-$t$, and so on. The use of this sort of flexible distributions for positive data helps you avoid trying to find the right transformation that makes your data normal, which can be very difficult to find or may not be available in closed form. Instead, you focus on appropriately modelling the features of your data.