Why is the Linearity of Expectation Used in This Equation?

[cappadonza]

Hi I'm going through some presentation material and i can't understand how the following has been derived

$$\sum^{n}_{j=1} \mathbb{E}[ ln(1 +K_{j})] = n \mathbb{E}[ln(1+K_{1})]$$

Could someone point me in the right direction on why this makes sense ?

Thanks

 

[mathman]

It only makes sense if all K_j have the same distribution, although it could hold (by accident) in more general situations.

 

[cappadonza]

ok suppose all $$K_{j}$$ have exactly the same distribution, I still can see why it makes sense. why does the following hold $$\sum^{n}_{j=1} \mathbb{E}[ ln(1 +K_{j})] = n \mathbb{E}[ln(1+K_{1})]$$

maybe there is something trivial here that I'm missing but i still can't see it

 

[statdad]

If all the $K_j$ have the same distribution, so do all of the [itex]\ln (1+K_j)[/tex], and this common distribution is the same as that of $\ln(1+K_1)$.<br /> <br /> If they have the same distribution, and if the expectations exist, then every term in the first sum is the same.[/itex]