Unit of a log normal probability density function How do I find the unit of a log-normal probability density function?
 A: The units of a probability density function (PDF) for a quantity $x$, are the inverse of the units of $x$. For example, if $x$ has units of length, then the PDF $P(x)$ has units of 1/length, so that the probability $P(x) dx$ is dimensionless.
The fact that the functional form of the PDF may be a log-normal distribution is not relevant to the dimensions of the PDF.
Note, that the acronym PDF can also refer to a probability distribution function, in the context of discrete random variables. Of course, a log-normal distribution is continuous so your question presumably refers to probability density functions. Nevertheless, to avoid confusion, as pointed out by @JohnDarby, in the context of discrete random variables, the probability distribution function is dimensionless, because it gives dimensionless probabilities for each possible outcome.
A: Consider any continuous random variable $V$ with the probability density function $p_V$. For $v$ any specific value of $V$, $p_{V}(v)$ is not a probability. The probability a specific value of $V$ is exactly $v$ is always zero and is meaningless; what is meaningful is the probability that $V$ is within $dv$ about $v$ and that probability is $p_V(v)dv$ which is always dimensionless regardless of the units of V as it must be since it is a probability.  So the units of $p_{v}$ are the inverse units of $V$.  See the earlier answer by @Andrew.
For a discrete random variable $R$ with probability density $p_{R}$, the probability that $R$ is the specific value $r$ is $p_{R}(r)$. $p_{R}$ for a discrete variable is the "probability mass function" sometimes called the "probability density function" and it always dimensionless regardless of the units of R as it must be since it is a probability.
A log-normal distribution is a special case. A log-normal distribution for the continuous random variable $X$ means that the logarithm of $X$ is normally distributed.  If $Y=ln(X)$ and Y is normally distributed, then the distribution for $X$ is a log-normal distribution.  The variable Y is dimensionless since it is the logarithm of a number. So, the probability density function $p_{Y}$ is also dimensionless for this special case. $X$ can have dimensions, so the units of $p_{X}$ are the inverse units of $X$.
