Law of the Unconscious Statistician

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Source: Lesson 24 LOTUS

Theorem (LOTUS): Let \(X\) be a random variable with p.m.f. \(f_X(x)\). Define \(Y=g(X)\) for some function \(g\). Then, \(E[Y]=E[g(X)]\) is \[ E[g(X)]=\sum_x g(x) \cdot f_X(x) \]

Theorem 24.1 allows us to calculate the expected value of \(Y=g(X)\), without first finding its distribution. Instead, we can just use the known distribution of \(X\).

This result is called the "Law of the Unconscious Statistician" because many people intuitively assume it is true. Remember that \(E[g(X)]\) represents the "average" value of \(g(X)\). To calculate the average value of \(g(X)\), it makes sense to take a weighted average of the possible values \(g(x)\), where the weights are the probabilities \(f_X(x)\).