# Find a function from a Recursive Formula

1. Mar 12, 2013

### loveinla

Hi,

I'd like to solve a function $$W_{t}^{x}$$ from a recursive formula below.

$$W_{t}^{x}=P_{t}^{x}*W_{t+1}^{x+1}+(1 - P_{t}^{x})*W_{t+1}^{x}$$,

where $$P_{t}^{x} = \frac{E[\theta^{x+1}*(1 - \theta)^{t-x}]}{E[\theta^{x}*(1-\theta)^{t-x}]}$$, $${\underset{t\to\infty, x\to 0}{lim}}W_{t}^{x} =0$$. Here $$E$$ stands for expectation.

Any suggestions? Thanks.

2. Mar 12, 2013

### Stephen Tashi

Expectation with respect to which variable? Over what range of integration?

3. Mar 12, 2013

### dodo

I'd imagine on the only free variable in the expression for P, namely $\theta$, and over $(-\infty,\infty)$. But you do well in asking instead of guessing like yours truly. :)

What worries me most is that, in the recurrence rule for W, the "x" does not get any smaller. So what are supposed to be the initial values?

4. Mar 12, 2013

### loveinla

Thanks, you are right, the expectation is with respect to $\theta$, which is on [0,1].

The initial value is indeed what I am trying to find.

In fact, it is a one-armed bandit problem with Bernoulli probability $\theta$, but $\theta$ is a random variable we are trying to figure out through the realization.

Any idea on how to determine the initial value $W^{0}_{0}$ or $W^{x}_{t}$?