# Complementary CDF for log-normal distributed function

I am reading a paper and trying to understand how authors have come up with the certain equation.

I understand the log-normal distribution and the relation to the normal distribution. But cannot understand the following simple derivation. (Another question is why have authors taken CCDF here?)

The following is the exact excerpt from the paper.

The shadowing effect is modeled by a log-normal random variable H. In other words, $10 \log H$ is a Gaussian r.v. with standard deviation $\sigma$. Let $f_H$ be the PDF (Probability Density Function) of $H$, and $F_H$ is the complementary CDF of $H$.

## 1 Answer

The complementary CDF is more convenient when talking about the tail behavior. Supposedly one can find out how the CCDF is used in the rest of the paper.

The "derivation" is simply plugging in the definition. Just note that reasonably one reads $\log\beta$ as the $10$-based logarithm therefore $\log \beta = \ln\beta\, / \ln10$.

Namely, by definition denote $U = \dfrac{10 \log \beta}{\sigma} = \dfrac{10 \ln \beta}{\sigma \ln 10}$ as the standard Normal random variable with mean $0$ and variance $1$, then you plugin the definition of the CCDF $Q(x) = \Pr\{ U > x\}$.