By Klotz J.H.

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T |H r H’s and y T ’s with the last outcome the final rth head. This sequence has probability pr q y y+r−1 such outcomes applying the binomial counting and there are y rule for sequences of two types of elements to the first r − 1 H’s and y T ’s . Thus y+r−1 pr q y for y = 0, 1, 2, . . , ∞ . P (Y = y) = y It gets its name from the coefficients in the negative binomial expansion (p + q)−r . The probabilities add to 1 since ∞ y=0 y+r−1 y pr q y = (p + q)−r = 1−r = 1 . 64 CHAPTER 3. RANDOM VARIABLES The special case where r = 1 is called the geometric distribution.

We then define pi P (A) = i: ei ∈A This definition satisfies the probability axioms. Axiom (i) holds since pi ≥ 0 and P (S) = 1, we have 0≤ i: pi = P (A) ≤ pi = P (S) = 1 . i: ei ∈S ei ∈A Axiom (ii) holds since we can break up the sums for disjoint events {Ai : i = 1, 2, . . , ∞ where Ai ∩ Aj = φ for i = j}. Note, for finite sample spaces, all but a finite number of the sets are the empty set φ. For these we use the convention that a sum over no outcomes is zero. P( ∞ pj = Ai ) = i=1 Sj : ej ∈ ∞ i=1 ∞ i=1 Ai pj i = ji : eji ∈Ai ∞ P (Ai ) .

Such a collection is a class of sets A with the following properties: (i). S ∈ A. (ii). If A ∈ A then Ac ∈ A. (iii). if Ai ∈ A for i = 1, 2, . . , ∞ then ∞ i=1 Ai ∈ A. Using (i) and (ii) we have φ = S c ∈ A. If Ai ∈ A for i = 1, 2, . . , ∞ then ∞ i=1 Ai ∈ A using DeMorgan’s rule and (ii),(iii) ∞ i=1 Ai = ( ∞ i=1 Aci )c ∈ A . 3. PROBABILITY Using the infinite sequence {Ai : i = 1, 2, . . , ∞} where Ai = φ for i = N + 1, N + 2, . . , ∞ we have using (iii) ∞ N Ai = i=1 i=1 Ai ∈ A . Then using this finite union, DeMorgan’s rule, and (ii),(iii) we have N N Ai = ( i=1 i=1 Aci )c ∈ A and finite unions and intersections belong to A as well as countably infinite unions and intersections.

### A computational approach to statistics by Klotz J.H.

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