Properties of Expected Value / Expectation
- 𝐄[𝑋 + 𝑌] = 𝐄[𝑋] + 𝐄[𝑌]
- 𝐄[𝑎𝑋] = 𝑎𝐄[𝑋]
- 𝐄[𝑐] = 𝑐
- 𝐄[𝑎𝑋 + 𝑏𝑌 + 𝑐] = 𝑎𝐄[𝑋] + 𝑏𝐄[𝑌] + 𝑐
- 𝐄[𝑋𝑌] = 𝐄[𝑋]𝐄[𝑌] # for independent 𝑋 and 𝑌
- 𝐄[𝑋𝑌] = 𝐶𝑜𝑣(𝑋,𝑌) + 𝐄[𝑋]𝐄[𝑌] # for non-independent 𝑋 and 𝑌, 𝐶𝑜𝑣(𝑋,𝑌) is the covariance
- 𝐄[𝑋2] = 𝑉𝑎𝑟(𝑋) + 𝐄[𝑋]2# derived from the second central moment