# Homework Help: Expected value of stationary independent increments.

1. Apr 7, 2010

### dionysian

1. The problem statement, all variables and given/known data
Let $$\{ X(t),t \ge 0\}$$ be a random process with stationary independent increments and assume that $$X(0)=0$$. Show that:

$$E[X(t)] = {\mu _1}t$$

2. Relevant equations

3. The attempt at a solution
I tried to work backwards by argueing that the mean between time intervals it equal to the product of the mean at t =1 and the time interval
$$\mu \Delta {t_1} + \mu \Delta {t_2} + \cdot \cdot \cdot + \mu \Delta {t_n} = \mu (\Delta {t_1} + \Delta {t_2} + \cdot \cdot \cdot + \Delta {t_n}) = \mu {t_n}$$

Allthough this gives me the write ansewr i dont feel its a very solid proof. Its seems there should be a straight forward way to show this with the diffenition of the expected value but i cant seem to see it.

Last edited: Apr 7, 2010
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