1


2

 Finite distribution take only
finitely many values.
 Examples of finite distributions:
 Bernoulli(1/2) Uniform
on {1,2,3,4,5,6}
 P(0)=1/2, P(1)=1/2. P(i) = 1/6.
 Number of heads in a coin flip.
Numbers on a fair die.

3


4

 An infinite discrete distribution on {0,1,2,…} is given by a sequence p_{0},p_{1},p_{2}
of probabilities s.t
 å_{i=0}^{1} p_{i} = 1 and p_{i}
¸ 0 for all i.
 Examples of infinite discrete distribution:
 Geometric (p = 1/6) Poisson (m = 1);
 P(n)=(5/6)^{n1} 1/6, n ¸ 1 P(n) = e^{1}/n!, n ¸ 0

5

 The infinite sum rule: If the event A is partitioned into A_{1},
A_{2},A_{3, }...
 A = A_{1}[A_{2}[A_{3}[ … and
 A_{i} Å A_{j} = Æ for all i ¹ j then
 P(A) = P(A_{1}) + P(A_{2}) + P(A_{3}) + …
 Ex: Let T = # rolls needed to produce a 6
 What is the probability that T > 10?
 What is the probability that T is even?
 What is the probability that T is finite?

6

 Let A be the event {six is
rolled sometime}.
 Then A can be partitioned into mutually exclusive subevents:
 A_{i}={the first six is on the i^{th} roll}

7

 P(T>10) = Σ_{i=11}( P(T=i) )
 P(T is even) = Σ_{i=1}( P(T=2i) )

8


9


10


11


12

 In the long run the average number of
 wins per game is (P(F) + P(M)) and the
 average number of games per win is the reciprocal.

13

 Recall that for a X = Geom(p) distribution,
 P(X = x) = p(1p)^{x1} and P(X ≥ x) = (1p)^{x1}
 So E(X) = Σ(1p)^{x1} by the tail sum formula.
 This is a geometric series , which we should all know sums to 1/(1 –
(1p)) = 1/p.
 What about Var(X)?

14


15


16

 Let W_{i} denote the waiting time after
 the (i1)st success until the i^{t}h success.

17

 Suppose each box of a particular brand of cereal contain 1 of n
different Simpsons sticker and the sticker in each box is equally likely
to be any one of the n, independently of what was found in the other
boxes.
 What is the expected number of boxes a collector must buy to get the
complete set of stickers?

18

 After getting k of the stickers, the additional number of boxes needed
to get a different sticker is a Geometric(p_{k})_{ }random
variable with
 p_{k} =(nk)/n .

19


20


21

