Finding expectation of the function of a sum of i.i.d. random variables

[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.