1. ## Tank Problem

A tank has some stochastic inflow of water, a water level and some deterministic outflow for different time steps. What I need is the probability density function for the water level in the tank at time step t+1, $\displaystyle T_{t+1}(n)$.

The probability density function for the inflow is given by $\displaystyle G_t(i)$ , the outflow by u and the tank level at time $\displaystyle t_0=n_0$. The inflow is assumed to be independent between the time steps.

Does anyone have some suggestions?

2. Originally Posted by Robert Hall
A tank has some stochastic inflow of water, a water level and some deterministic outflow for different time steps. What I need is the probability density function for the water level in the tank at time step t+1, $\displaystyle T_{t+1}(n)$.

The probability density function for the inflow is given by $\displaystyle G_t(i)$ , the outflow by u and the tank level at time $\displaystyle t_0=n_0$. The inflow is assumed to be independent between the time steps.

Does anyone have some suggestions?
What happens when there is insufficent water in the tank to support the outflow?

If we allow a water debt when there is insufficient to support the outflow (so the outflow is maintained ) then the water content of the tank at epoc $\displaystyle t$ is the original level plus the inflows minus the outflows. Thus we would only need to know the distribution of the RV representing the total inflow up to epoc $\displaystyle t$:

$\displaystyle C_t=\sum_{i=1}^t I_i$

where $\displaystyle I_i$ is the RV representing the inflow between $\displaystyle t=i-1$ and $\displaystyle t=i$

CB

3. I was thinking something like this:
Say that X is the water level at time t=1 and that Y is the water level at time t=0. Then

$\displaystyle f_{X \mid Y=n_0}(n) = \frac{f_{X,Y}(n,n_o)}{f_Y(n_0)}$

$\displaystyle f_{X,Y}(n,n_o)=f_Y(n_0) f_{X \mid Y=n_0}(n)$

$\displaystyle f_{X}(n)=\int f_Y(n_0) f_{X \mid Y=n_0}(n) dn_0$

Where $\displaystyle f_{X}(n)$ is the probability density function for the water level at time t+1.

$\displaystyle f_Y(n_0)$ should be given by
$\displaystyle f_Y(n_0) = G_0(i)-u+n_0$

But what is $\displaystyle f_{X \mid Y=n_0}(n) ?$