Major issue to handle this proof

[nicekiller231]

Good morning.

My problem is as follow:

I have an event assuming A. The probability that A occurs at time t is: \(\displaystyle p(t)= e^{-bt}*|\sin(at)|\). Where a,b are positive parameters.

We divide the time in small step times let's say \delta t= 0.125, Then, we count how many time A occur for $t \in [0, \infty]$. Good to notice that $$\lim_{{t}\to{\infty}}p(t)=0$$

So my problem is to study the number of occurrence of A in the variation of the parameter a and b.
Which I can prove mathematically that for a lower value of b, A occurs more often and for the bigger value of a, A occurs more frequently.

I hope I was clear.
If anyone has any suggestion or idea about how could we do that.
Thank you.

 

[HallsofIvy]

As far as b is concerned, noting that $$e^{-bt}$$, for fixed t, is a decreasing function of b (for example, the derivative with respect to b, $$-te^{-bt}$$, is negative) is sufficient. As for a, I don't believe it is necessarily true that "for the bigger value of a, A occurs more frequently."

 

[nicekiller231]

As far as b is concerned, noting that $$e^{-bt}$$, for fixed t, is a decreasing function of b (for example, the derivative with respect to b, $$-te^{-bt}$$, is negative) is sufficient. As for a, I don't believe it is necessarily true that "for the bigger value of a, A occurs more frequently."

actually, It is the case for higher values of $a$ A occur more often. However, it is obvious for the parameter b.
But in my case I looking for a proof to that.

This is my way but I not sure if it is the right way to do it
$A(t)$ is $p(t)$, $b$ is $ \beta$ and a is $\omega $
we estimate $\phi(\omega , \beta)$, which is the primitive of a function $A(t)$ representing the individual's attraction to a rumor from the initial time (of hearing the rumor) until the loss of interest by that individual.
We prove that $\phi(\omega,\beta)$ is proportional to $\omega$ and inversely proportional to $\beta$.
\begin{align}
\phi(\omega,\beta)&=\int_0^\infty A(t) dt \\
&=\int_0^\infty A_{int} e^{-\beta t} |\sin(\omega t)| dt\qquad\mbox{for}\quad\beta >0
\label{equ:eq1}
\end{align}

Knowing that
\begin{equation*}
|\sin (\omega t)|=
\left\lbrace
\begin{array}{ll}
\sin (\omega t),\quad t \in [2k \pi/\omega,(2k+1)\pi/\omega]\\
-\sin (\omega t),\quad t \in [(2k+1)\pi/\omega, 2k\pi/\omega] \\

\end{array}
\right.
\quad \mbox{for } \omega \neq 0
\label{equ:eq2}
\end{equation*}
Knowing the expression of $|\sin(\omega t)|$, we obtain for $ \beta > 0$ and $\omega \neq 0$
\begin{align}
\phi(\omega,\beta)&= A_{int} \sum_{k=0}^{\infty} \int_{k \pi/\omega}^{(k+1)\pi/\omega} (-1)^k e^{-\beta t} \sin(\omega t) dt \\
& =A_{int} \sum_{k=0}^{\infty} \frac{\omega e^\frac{-\beta k \pi}{\omega}}{\beta^2+\omega^2}(1+e^\frac{-\beta \pi}{\omega}) \\
& =\frac{A_{int}\omega \left( 1+e^\frac{-\beta \pi}{\omega}\right) }{(\beta^2+\omega^2)(1-e^\frac{-\beta \pi}{\omega})}
\end{align}
Then we study $ \frac{\partial \phi(\omega,\beta) }{\partial \omega}$ and $ \frac{\partial \phi(\omega,\beta) }{\partial \beta}$ to highlight the influence of $\omega $ and $\beta$:
\begin{align}
\label{equ:eq8}
\phi_{\omega} &= - \frac{A_{int}\left(\left(\omega^3-\beta^2\omega\right)\mathrm{e}^\frac{2{\pi}\beta}{\omega}+\left(-2{\pi}\beta \omega^2-2{\pi}\beta^3\right)\mathrm{e}^\frac{{\pi}\beta}{\omega}-\omega^3+\beta^2\omega\right)}{\omega\left(\omega^2+\beta^2\right)^2\left(\mathrm{e}^\frac{{\pi}\beta}{w}-1\right)^2}\\
\label{equ:eq9}
\phi_{\beta} &= -\frac{2A_{int}\left(\omega\beta\mathrm{e}^\frac{2{\pi}\beta}{\omega}+\left({\pi}\beta^2+{\pi}\omega^2\right)\mathrm{e}^\frac{{\pi}\beta}{\omega}-\omega \beta\right)}{\left(\beta^2+\omega^2\right)^2\left(\mathrm{e}^\frac{{\pi}\beta}{\omega}-1\right)^2}
\end{align}The study of $ \frac{\partial \phi(\omega,\beta) }{\partial \omega}$ and $ \frac{\partial \phi(\omega,\beta) }{\partial \beta}$ shows that: First,
$\phi_\omega$ is positive for $\omega >0$ and $ \beta >0 $, from which we can conclude that $\phi(\omega,\beta)$ is proportional to $\omega$ in this interval.
Second, $\phi_\beta$ is a negative for $\beta > 0 $ and $ \omega >0$, from which we can conclude that $\phi(\omega,\beta)$ is inversely proportional to $\beta$ in the same interval.