# Failure criteria for a collection of independent evolving discrete random variables

I have built a computer model that contains a collection of independent discrete random variables. They each have values of between $0$ and $k$ where $k$ is between $4$ and $15$ and is constant for each variable. These variables initially start at a certain value (different for each) with a probability of $1$ and evolve probabilistically in discrete time steps based on another discrete random variable.

So, in symbols (this is my first try at this and my math notation is rather rusty so any feedback would be appreciated, if anyone can tell me why I get $t-1$ when I put in $t+1$ in the second equation that would be good)

$$\mathcal{T}_{(t)}=\sum_{i=1}^{35}\mathcal{X}_{i(t)}$$

and

$$\mathcal{X}_{i(t+1)}=\mathcal{X}_{i(t)}\times\mathcal{Y}_{i(t)}$$

where

$X_{i(t)}$ is the $i$th random variable at time $t$ and has a range of 0 to $k$

$\mathcal{T}_{(t)}$ is the a discrete rendom variable total at time $t$ and has a range of $0$ to $290$.

$Y_{i(t)}$ is another (well known but changeable) random variable that acts on $X_{i(t)}$ to evolve the solution

The model is working quite nicely and each variable happily evolves in ignorance of all the others.

What I am interested in is the probability of the sum of these random variables falling below a threshold value of $90$ for time $t$. In the real world these variables represent the status of a game and if the total falls below the threshold then one of the players wins and the game is over.

In symbols

$$P(\mathcal{T_{(t)}})\lt90$$

If I simply continue the evolution, then there will be some states at $t+1$ with $\mathcal{T_{(t)}}\ge90$ that were derived from a state at $m$ with $\mathcal{T_{(t)}}\lt90$. In a naive treatment these would be considered non-winning states even though they derived from a state where the game ended the turn before.

I think that the evolution of each variable needs to take into account the sum of all the other variables (or derive it from the sum of all the variables less its own contribution) but I am not sure how to proceed.

Any guidance would be appreciated.

My thinking on this so far:

At each time $t$ calculate each

$$S_{j(t)}=\sum_{i=1}_{i{\neq}j}^{35}\mathcal{X}_{i(t)}$$

and

$$P_j(P(\mathcal{T_{(t)}})\lt90|S_{j(t)})$$

for each possible value of $X_{j(t)}$ and modify $X_{j(t)}$ by this probability so that

$$P(X_{i(t)})=1-P_j(P(\mathcal{T_{(t)}})\lt90|S_{j(t)})$$

this would be the distribution given that the game had not ended and could be used to move on to $t=t+1$. So we can then calculate

$$P(E_{(t)})=P(E_{(t-1)})+P(\mathcal{T_{(t)}})\lt90$$

where $E_{(t)}$ is the event end of the game at or before time $t$

Is this a sound approach?

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How do these r.v. evolve precisely? This is the key for the computation of the probability that the sum is below the threshold at some moment $m$ –  Ilya Jan 11 '13 at 22:31
Formatting pays more than bounties. –  leonbloy Jan 11 '13 at 22:56
@Ilya I have redrafted the problem. I dont think the exact mechanics of how it moves from time t to t+1 are important but have a look and see if you need fuurther info –  Dale M Jan 12 '13 at 1:19
@leonbloy - This is my first attempt, I have had a go at redrafting the question. Would you have another look please? –  Dale M Jan 12 '13 at 1:20