I have a random function f(n) which takes the values +/- 1 with equal probability.(adsbygoogle = window.adsbygoogle || []).push({});

Let the variable X take the sum of the values of f(n) after n steps. Then I can write,

[tex]X(T) = \sum_{n=0}^T f(n)[/tex] where T = 0,1,2,... and X(0) = 0.

And I can write the expectation of X as,

[tex]<X> = < \sum_{n=0}^T f(n) > = \sum_{n=0}^T <f(n) > = 0[/tex] since <f(n)> = 0 (by definition).

My question:

if, instead [tex]X(n) = \sum_{k=0}^n e^{-\mu k} f(n-k)[/tex] then, for the expectaion of X, can I write,

[tex]< X(n)> = \sum_{k=0}^n e^{-\mu k} < f(n-k) >[/tex]

or even,

[tex]< X(n)> = \sum_{k=0}^n < e^{-\mu k}> < f(n-k) >[/tex]

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

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# Random walk:expectation

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