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Mathematics
Set Theory, Logic, Probability, Statistics
Finding expectation of the function of a sum
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[QUOTE="Office_Shredder, post: 6866056, member: 53426"] Here's what I think the solution is to my problem. I'm going to add two characters to make the notation more clear. ##E(\sin(S_n)|S_n=x)##. The expected value of the sine, given ##S_n## begs the question, given ##S_n## what? Given ##S_n## means given the realized outcome of ##S_n## from the random trial. I've decided to call that outcome ##x##. So if someone tells you they sampled all the random variables and this time ##S_n=3## what is he expected value of ##\sin(S_n)##? Hopefully the answer is obvious, ##\sin(3)##. It's the only possible value of ##\sin(S_n)##. If we want to be very formal we could write this as ##E(\sin(S_n)|S_n=3)=\sum_y \sin(y) P(S_n=y | S_n=3)##. (In fact this is what I know as the definition of this notation) Obviously ##P(S_n=y|S_n=3)## is 0 except for ##y=3## where it's 1. There's nothing special about 3, in general ##E(\sin(S_n) | S_n=x)=\sin(x)##. Now in your original problem, being told ##S_n^2=16##, to pick a random possible number, doesn't restrict ##S_n## to only a single value. But there aren't that many choices for what it can be. If looks in your last post like you were hitting on this, but I don't know why you wrote ##\sin(S_n=0)## for example, or what that even means. Also, ##S_n^2## is very unlikely to actually equal ##n## exactly. [/QUOTE]
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Finding expectation of the function of a sum
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