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I am trying to understand a research article which states the following,

$h_{t,u}^v \sim exp(1)$ denotes the exponentially distributed fading power from transmitter $t$ to the receiver $u$ over Rayleigh fading channel.

Can anyone help me to understand what it means? Also, what does $exp(1)$ mean here?

For detail relevant to the question image is attached below.

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PS. Please ignore the other terms. I just want to inquire about $h \sim exp(1)$ here. In case someone is curious, $P_M$ is Power by the transmitter, $l_{\alpha N}(m,u)$ is the distance-dependent fading for the channel and $u$ is the receiver.

SJa
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  • It's an exponentially distributed random variable with parameter $\lambda=1$. See https://en.wikipedia.org/wiki/Exponential_distribution. – symplectomorphic Jan 10 '18 at 02:50
  • Yes, I understand this thing. What I mean to say what does this mean for the wireless channel? What will happen to wireless channel if $\lambda = 1$ – SJa Jan 10 '18 at 02:54
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    You seem not to understand, because you asked "what does $exp(1)$ mean here?" The notation $X\sim\text{exp}(1)$ means that $X$ is a random variable that has the exponential distribution with parameter 1. Thus $X$ randomly takes on a value in the interval $[0,\infty)$, but small values are more likely than large values. The parameter $1$ means, for example, that the distribution has a mean and variance of $1$. – symplectomorphic Jan 10 '18 at 03:01
  • Make sense. Understood! Thanks :) Why don't you put this in the answer and I can accept it? – SJa Jan 10 '18 at 03:25

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The notation $X\sim\text{exp}(1)$ means that $X$ is a random variable that has the exponential distribution with parameter $1$. Thus $X$ randomly takes on a value in the interval $[0,\infty)$, but small values are more likely than large values.

The parameter $\lambda$ governs the exact shape of the distribution. For a distribution with parameter $\lambda$, the mean is $\lambda^{-1}$ and the variance is $\lambda^{-2}$.

symplectomorphic
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