Stat 5101 Lecture Notes - School of Statistics

2.5. PROBABILITY THEORY AS LINEAR ALGEBRA 61A little section about mathematics is invariant under changes of notation.We often write (2.40) in different notation. If X is a random variable withdensity f X having domain S (the range of possible values of X), thenE{g(X)} = ∑ g(x)f X (x). (2.42)x∈SNote that (2.42) is exactly the same as (2.40) except for purely notational differences.The special case where g is the identity functionE(X) = ∑ x∈Sxf X (x) (2.43)is of some interest. Lindgren takes (2.43) as the definition of expectation. For usit is a trivial special case of the more general formula (2.42), which in turn is nota definition but a theorem (Theorem 2.27). For us the definition of expectationis “an operator satisfying the axioms.”Example 2.5.1 (The Binomial Distribution).Recall the binomial distribution (Section B.1.2 of Appendix B) having density( nf(x) = px)x (1 − p) n−x , x =0,...,n.We want to calculate E(X). By the formulas in the preceding discussionE(X) ==n∑xf(x)x=0n∑k=0( nk pk)k (1 − p) n−kn∑ n!= kk!(n − k)! pk (1 − p) n−kk=0n∑ n!=(k − 1)!(n − k)! pk (1 − p) n−kk=1n∑ (n − 1)!= np(k − 1)!(n − k)! pk−1 (1 − p) n−kk=1n∑( ) n − 1= npp k−1 (1 − p) n−kk − 1k=1n−1∑( ) n − 1= npp m (1 − p) n−1−mmm=0•Going from line 1 to line 2 we just plugged in the definition of f(x) andchanged the dummy variable of summation from x to k.