We may think of the expected value as an operator mapping a random variable X into its expected value μ = E[X]. The expectation operator enjoys two very useful properties.
PROPERTY 6.6 (Linearity of expectation 1) Given a random variable X with expected value E[X], we have
for any numbers α and β.
This property is fairly easy to prove:
Informally, the property provides us with a quick rule for manipulating expectation as an operator, stating that numbers can be “taken outside” the expectation. The next property is a bit less trivial, as it involves the sum of multiple random variables.
PROPERTY 6.7 (Linearity of expectation 2) Given m random variables Xi, i = 1,…, m, we have
A proof of this property is a bit involved, as it requires some tedious algebra, and it is omitted. What makes this property conceptually not trivial is that when dealing with multiple random variables, some care might be needed as they may have different distributions, and their mutual relationships may be quite complicated. Rather surprisingly, Property 6.7 states that the expected value of a sum of random variables is always the sum of their expected values.5
Taken together, the two properties state that expectation is a linear operator, in the sense that the expected value of a linear combination of random variables, is just the linear combination of their expected values:6
for arbitrary coefficients αi, i = 1,…, m.
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