Calculating Expectation for a Random Walk with a Time-Varying Function

[Steve10]

I have a random function f(n) which takes the values +/- 1 with equal probability.
Let the variable X take the sum of the values of f(n) after n steps. Then I can write,
X(T) = \sum_{n=0}^T f(n) where T = 0,1,2,... and X(0) = 0.
And I can write the expectation of X as,
&lt;X&gt; = &lt; \sum_{n=0}^T f(n) &gt; = \sum_{n=0}^T &lt;f(n) &gt; = 0 since <f(n)> = 0 (by definition).
My question:
if, instead X(n) = \sum_{k=0}^n e^{-\mu k} f(n-k) then, for the expectaion of X, can I write,
&lt; X(n)&gt; = \sum_{k=0}^n e^{-\mu k} &lt; f(n-k) &gt;
or even,
&lt; X(n)&gt; = \sum_{k=0}^n &lt; e^{-\mu k}&gt; &lt; f(n-k) &gt;
If I can do either of the above then can I also use <f(n)> = 0 to say that <X(n)> = 0, which I'm pretty sure it would have to be to answer the rest of my question.

TIA

 

[mathman]

You left out one important definition - what is mu? Is it a random variable or is it a constant? If it is random, how does it relate to f(n)? If it is constant or independent of f(n), then your conclusion is correct.

 

[Steve10]

Thanks, I should have mentined that about mu. It is a constant value.
So I was correct after all.
I guess that's just as well, since I'm already typing up my results based on that conclusion!
Many thanks for the confirmation.