So I have come across a question asked by my peers.
Define: $$g:=\sqrt{E[|y_r(t)|^2]}$$
Given that $$y_r(t)=\sqrt{t}\cdot h+b+k+c,$$ where $h$, $b$, $k$, and $c$ are independent random variables. The variance of $k$ and $c$ are $\sigma_k$ and $\sigma_c$, respectively. (Note that $h$ and $b$ are 1 kind of random variable, they represent the communication channel coefficient. On the other hand, $k$ and $c$ are another kind of random variable; they represent the additive white gaussian noise (AWGN)).
So what is $g$ in this case? The answer key gives: $g=\sqrt{t |h|^2 + \sigma_k+|b|^2+\sigma_c}$ .
I am confused, over which random variables does it apply the expectation? Does the mean of $k$ and $c$ must be zero in order for this expression to be true?
How come $k$ got converted to $\sigma_k$ but not $h$ and $b$?
If the modulus sign means a normal mathematical modulus, then I think the key has to be wrong. (I don't see the reason why you need it, as square will take care of the sign anyway, unless we are working with complex variables. ) If we assume that h,b,k,c are independent variables in very general terms.
$$ E[y^2]= E[th^2 + b^2+k^2+c^2 +2bk+2bc+2bh \sqrt t +2kc+2ch \sqrt t+2kh \sqrt t]=E[th^2]+E[b^2]+... $$ For any random variable x $$ E[x^2]=\sigma_x^2+E[x]^2 $$ If x,y are independent. $$ E[xy]=E[x]E[y] $$ Therefore we have: $$ E[y^2]=\sigma_k^2+\sigma_c^2+E[k]^2+E[c]^2+2E[k]E[c]+... $$ Continue for all other random variables.
Here we see that first of all we get $\sigma_x^2$ instead of $\sigma_x$ which tells me that the key has to be incorrect.
To get as close as mathematically possible to the key we need to assume that $E[k] = 0$ $E[c] = 0$ and that b and h are not random variables. (Argument against that fact is that in the key g is in functional relation to them which leaves g itself a random variable and not a statistic, not a number.)
Then we will get $$ g=\sqrt {\sigma_c^2+\sigma_k^2 +th^2+b^2+2\sqrt thb} $$