I have a real valued number y_t. At each time
I have a real valued number . At each time step t, is multiplied by with probability p and multiplied by with probability . What is the expected value of ? What is the variance?
Answer & Explanation
It is quite simple if you use independence in a more direct way and the method works for any distribution. Assuming that where the are i.i.d. factors. By independence:
valid for any distribution. In the specific (Bernoulli) example:
again valid for any distribution. In our case: . In particular, and you may carry on from there, in order to calculate limits etc... (e.g. gives a nice limit). The actual distribution of is in general quite complicated.
Assuming independence and regarding the expectation:
For the sake of simplicity, let . The value of our variable, at the moment, is
(Where k denotes the number of multiplications by .)
The expected value is, then
because of the binomial theorem.
Unfortunately I could not find the trick for the calculation of the expectation of the square of the same random variable...
Following up zoli's answer, I think can be found the same way, with the Binomial theorem yielding (assuming )
So the variance is . I don't know if it can be simplified any further.
Quick check: as