2.3 Functions of discrete random variables 29 10−5. It may be easier (and not too inaccurate) to use (2.20) rather than (2.19) to calculate probabilities. In this case, λ = np = 10 and so, for example, P(Sn = 10) ≈

1 10!

(10e−1)10 ≈ 0.125. △

vExample 2.21 Suppose that we toss the coin of the previous example until the first head turns up, and then we stop. The sample space now is

 = {H, TH, T2H, . . . } ∪ {T∞}, where TkH represents the outcome of k tails followed by a head, and T∞ represents an infinite sequence of tails with no head. As before, F is the set of all subsets of , and P is given by the observation that P(TkH) = pqk for k = 0, 1, 2 . . . , P(T∞) =

(

1 if p = 0, 0 if p > 0.