Tips on Calculating Expectations
Let `X` be a random variable.
`E[a+bX] = a+b E[X]`
`E[a * g_1(X) + b*g_2(X) + c] = a E[g_1(X)] + b E[g_2(X)] + c`
If `g_1(x) <= g_2(x)` for all `x` in the support of the random variable `X`, and the expectations `E[g_1(X)]` and `E[g_2(X)]` exist, then `E[g_1(X)] <= E[g_2(X)]`.
Jensen's Inequality: see Planetmath.