A Rayleigh-faded channel is often written as

which is short-hand for saying that 

where is a normal distribution with mean and variance .

We have in the case of .

The magnitude of the channel follows a Rayleigh distribution where whose PDF is

or, taking , is equivalent to

Note that when , .

Sometimes the Rayleigh distribution is parameterized by . Other times it is parameterized by .

The distribution of follows an exponential distribution with PDF

where (when ) and whose mean is , implying that .

In other words, the average channel power is 1. This is convenient for abstracting the large-scale path loss (e.g., Friis) and large-scale fading (e.g., log-normal shadowing) from the small-scale fading distribution (e.g., Rayleigh fading).

One could also define use to write the PDF as

Note that can also be written as having distribution

where now it’s clear that is the average power of (i.e., its variance is its average power since is zero mean).