Given an Exponentially Distributed Random Variable $X\sim \exp(1)$, I need to find $\mathbb{E}[P_v]$, where $P_v$ is given as:$$ P_v=
\left\{
\begin{array}{ll}
a\left(\frac{b}{1+\exp\left(-\bar \mu\frac{P_s X}{r^\alpha}+\varphi\right)}-1\right), & \text{if}\ \frac{P_s X}{r^\alpha}\geq P_a,\\
0...
I did not include the whole intermediate steps in the above solution which may cause confusion. Hence those steps are now presented below.
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**My solution**
\begin{multline}
\Phi = \mathbb{P}\left[ a\left(\frac{b}{1+\exp\left(-\bar \mu\frac{P_s X}{r^\alpha}+\varphi\right)}-1\right) \geq...
Given two i.i.d. random variables $X,Y$, such that $X\sim \exp(1), Y \sim \exp(1)$. I am looking for the probability $\Phi$. However, the analytical solution that I have got does not match with my simulation. I am presenting it here with the hope that someone with rectifies my mistake.
...
Is the following solution correct for the above question? If it is OK, then I have found the solution. But I will really appreciate if someone can let me know if the following method is correct.
$$\mathbb{E}[g[x]] = \int_Q^\infty g(x). f(x) dx$$
$$\mathbb{E}[g[x]]=...
Given that $X$ is exponentially distributed continuous random variable $X\sim \exp(1)$ and $g(x)$ is as below. How can I find the Expectectaion of $g(x)$ for the condition that $x\geq Q$, i.e. $\mathbb{E}[g(x)\ | \ x\geq Q]$.
$$g(x) = \frac{A}{\exp(-bQ+c)}\Big(\frac{1 + \exp(-bQ+c)}{1 +...
Let $a,b,c, \tau$ be positive constants and $x$ is an exponentially distributed variable with parameter $\lambda = 1$, i.e. $x\sim\exp(1)$.
\begin{equation}
E = \tau\Big[a\frac{1+a}{1+e^{-bx+c}} - 1 \Big]^+
\end{equation}
where $[z]^+ = \max(z,0)$
How can I find
The PDF for $E$
The...
I have an Energy harvesting expression something like the following
$R = \tau B \log\Big(1 + \frac{E h^2}{\tau r^\alpha\sigma^2} \Big)$
$E = \tau(2^{R/\tau B}-1 )\frac{r^\alpha\sigma^2}{h^2}$
Let all constant terms as $a$ to simplify the expression into : $E = a\frac{1}{h^2}$
$E$ is a random...
The following is the mathematical expression for my model's rate expression. Variables $x,y$ are the controlling parameter, while the rest are positive constants.
$$\max_{x,y} \ ax + by^3 \ (s.t. \ 0\leq x \leq 1,\ 0\leq y\leq1)$$
Can I mathematically say that it is a convex problem within...