Question

Question:

(a)What does the linearity of expectations of random variables mean?

(b)How can the linearity of expectations help us find the expected number of people who receive the correct hat when a hatcheck person returns hats at random?

Short answer

Answer

(a)\(X\) and \(Y\) are two different random variables with \(E(X + Y) = E(X) + E(Y)\).

(b) The resultant answer is the expectation of one person getting his hat back instead of all of them, and hence it simplifies the problem.

Step-by-step solution

Given data

The given data is a hatcheck person returns hats at random.

Concept of Linearity of expectations

(a)

It means that if:

\(X\) and \(Y\) are two different random variables then we have \(E(X + Y) = E(X) + E(Y)\).

Simplify the problem

(b)

It helps us because now we can divide the problem into \(n\) random variables \({X_1}, \ldots ,{X_n}\) with \({X_i}\) being 1 if the \(i - th\) person gets his hat back then the expected number of hats becomes \(n \times E\left( {{X_i}} \right)\) for any \(i\). So, we only need to calculate the expectation of one person getting his hat back instead of all of them, and hence it simplifies the problem.