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I am trying to study probability theory by myself. I asked myself a question about expectation, and I have no idea how to solve it. Please help me with solution and my understanding. The question I asked is following: Let $a, b, c, d$ be non-negative natural numbers, say from the set $\{0, 1, ..., n\}.$ Calculate the expectation $$ E\left( (-1)^{a+b+c+d}\right). $$

Thank you for your help.

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You should say something about the probability distribution of your four random variables $a,b,c,d$. –  Michael Hardy Feb 29 '12 at 3:32

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$(-1)^{a+b+c+d} = (-1)^a (-1)^b (-1)^c (-1)^d$. Assuming $a,b,c,d$ are independent (which you didn't say but is probably what is meant), the expected value of the product is the product of the expected values. $(-1)^a$ is $-1$ if $a$ is odd and $1$ if $a$ is even. So if the values $0,1,\ldots,n$ are all equally likely (which again you didn't say but you probably meant), it all comes down to counting how many odd and even numbers there are in $0,1,\ldots,n$.

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...and that depends on whether $n$ is odd or even. If $n$ is odd, then there are equally many odd and even numbers, so $E((-1)^a)$ would be $0$. If $n$ is even, it's a bit more complicated. –  Michael Hardy Feb 29 '12 at 3:34

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