Suppose the rate of interest $i$ on an investment is $20 \%$ with probability $0.1$, $5 \%$ with probability $0.4$,$2 \%$ with probability $0.4$, and $-10 \%$ with probability $0.1$.
How would you find $E[1+i]$?
Is $E[1+i]=1+E[i]$?
If so, why is $0.1(1+0.2)+0.4(1+0.05)+0.4(1+0.02)+0.1(1-0.1) \neq 1+0.1(0.2)+0.4(0.05)+0.4(0.02)+0.1(-0.1)$?