- Thread starter
- #1

Suppose we want to calculate the probability that we have transitions $C\to O\to C$

in the time interval $[0,\Delta t]$ where $C,O$ stand for different states. (we start in state $C$) And we have $T^+ \sim \exp(\lambda _+)$ and $T^- \sim \exp(\lambda_-)$ where $T^+$ is the time between a transition from $C\to O$, and $T^-$ the time between a transition $O\to C$. I want to show that the probability is of order $(\Delta t)^2$. So we need not know the probability explicitly. But suppose we try to do this explicity...

So i guess we need a triple integral for this...if we want to do this explicitly, we want to integrate over all $0<s_1<s_2\leq \Delta t$ so that

$P(T^+\leq s_1)$,

$P(T^-\leq s_2-s_1)$,

$P(T^+>\Delta t-s_1-s_2)$ (since we want after $C\to O \to C$ that it doesn't flip back to state $O$ within $[0,\Delta t]$)

And we simply assume independance...so I guess we can multiply the density functions

and make some triple integral with the correct boundaries. But i'm really confusing myself with this...

using the probability densities we must get some integral $\int\int\int ... ds_1ds_2dy$

Can someone see through my confusion, and point me in the right direction? I guess i'm not such a big calculus-star.

Thanks

---------- Post added at 01:50 PM ---------- Previous post was at 01:39 PM ----------

Could it be that this is the correct integral? $\int_0^{\Delta t}p_+(s_1)\left(\int_{s_1}^{\Delta t}p_-(s_2)\left(\int _{\Delta t-s_1-s_2}^{\infty}p_+(x)dx\right)ds_2\right)ds_1$

---------- Post added at 02:55 PM ---------- Previous post was at 01:50 PM ----------

edit 2: sorry...think i solved itHoi. I want to prove the following:

Suppose we want to calculate the probability that we have transitions $C\to O\to C$

in the time interval $[0,\Delta t]$ where $C,O$ stand for different states. (we start in state $C$) And we have $T^+ \sim \exp(\lambda _+)$ and $T^- \sim \exp(\lambda_-)$ where $T^+$ is the time between a transition from $C\to O$, and $T^-$ the time between a transition $O\to C$. I want to show that the probability is of order $(\Delta t)^2$. So we need not know the probability explicitly. But suppose we try to do this explicity...

So i guess we need a triple integral for this...if we want to do this explicitly, we want to integrate over all $0<s_1<s_2\leq \Delta t$ so that

$T^+\leq s_1$,

$T^-\leq s_2-s_1$,

$T^+>\Delta t-s_1-s_2$ (since we want after $C\to O \to C$ that it doesn't flip back to state $O$ within $[0,\Delta t]$)

And we simply assume independance...so I guess we can multiply the density functions

and make some triple integral with the correct boundaries. But i'm really confusing myself with this...

using the probability densities we must get some integral $\int\int\int ... ds_1ds_2dy$

Can someone see through my confusion, and point me in the right direction? I guess i'm not such a big calculus-star.

Thanks

---------- Post added at 01:50 PM ---------- Previous post was at 01:39 PM ----------

Could it be that this is the correct integral? $\int_0^{\Delta t}p_+(s_1)\left(\int_{s_1}^{\Delta t}p_-(s_2)\left(\int _{\Delta t-s_1-s_2}^{\infty}p_+(x)dx\right)ds_2\right)ds_1$