Finding expectation of the function of a sum

[WMDhamnekar]

TL;DR Summary

Finding expectation of the function of a sum of i.i.d. random variables given the 2nd moment of sum of i. i. d. random variables.

My answer:

Is the above answer correct?

 

[mathman]

##S_n$$ has a discrete distribution. How did you get continuous?

 

[WMDhamnekar]

Oh, I am sorry. You are correct. So, if ## S_n## has a discrete distribution, my answer would be
* ##S_n## is a random walk with symmetric steps, so ##E(S_n) = 0##
* ##S_n^2## is the sum of n independent random variables taking values 1 and -1 with equal probability, so ##E(S_n^2) = n##
* By conditional expectation, ##E(\sin{S_n} | S_n^2) = E(\sin{S_n} | n) = \frac{E(\sin{S_n})}{E(n)} = \frac{0}{n} = 0##.

 

[Office_Shredder]

I don't think you're handling the conditional part right. Your final answer is supposed to be a function of ##S_n^2## - given a specific value that it ends up being, what is the expected value of the sine?

 

[WMDhamnekar]

I don't think you're handling the conditional part right. Your final answer is supposed to be a function of ##S_n^2## - given a specific value that it ends up being, what is the expected value of the sine?

I edited my answer to this question. Does it look now correct?

 

[Office_Shredder]

No, I still think you have written meaningless notation.

Try to compute ##E(\sin(S_n) | S_n)## what does this even mean?

 

[WMDhamnekar]

No, I still think you have written meaningless notation.

Try to compute ##E(\sin(S_n) | S_n)## what does this even mean?

##S_n= \pm\sqrt{n}## with equal probability ##\frac12 \therefore E(\sin{(S_n)}|S^2_n)=\sin{(S_n=0)}\times \frac12 - \sin{(S_n=0)}\times \frac12 =0##

 

[Office_Shredder]

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.