Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation

Introduction to Probability Theory ‧3-1‧

Speaker: Chuang-Chieh LinAdvisor: Professor Maw-Shang Chang

National Chung Cheng University

Dept. CSIE, Computation Theory Laboratory

January 25, 2006

- Preliminaries for Randomized Algorithms

Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 22

Outline

• Chapter 3: Discrete random variables– Bernoulli and binomial distributions

– Geometric distribution

– Negative binomial distribution


Bernoulli trials ( 伯努利試驗 )

• A Bernoulli trial is an experiment with two different possible outcomes, labeled successsuccess and failurefailure. The sample space for a single Bernoulli trial is defined as T = {s, f}, where s represents the outcome success and f represents the outcome failure.


Bernoulli random variable

• If an experiment consists of a single Bernoulli trial with parameter p (so that P({s}) = p, and we denote q = 1 – p) and we let X be the number of successes to occur, then X is called a Bernoulli random variable with parameter p.

• Its probability function is very simple:

otherwise ,0

1,0for ,)(

1 xqpxp

xx

X


Bernoulli random variable (contd.)

• Mean and variance for a Bernoulli random variable X with parameter p:

pqppppXX

XpppX

X

XX

)1(])[E(][E

][E)1(1)0(0][E2222

2


• Many experiments can be modeled as a sequence of independent Bernoulli trials.

• For example,– Ten scratch-off lottery tickets are purchased; each ticket

either will or will not win some prize, where p is the probability of a success occurring for each.

– Each of 100 patients with the same affliction is given medication A ; each patient will either be cured or not, with the same success probability p.


Binomial random variable( 二項隨機變數 )

• If Y is the number of success to occur in n repeated, independent Bernoulli trials, each with probability of success p, then Y is a binomial random variable with parameter n and p. The range for Y is RY = {0, 1, 2,…, n}, and its probability function is

where q = 1 – p

otherwise. ,0

.for ,)( Y

yny

Y

Ryqpy

nyp


• 假設老王買了 10 張刮刮樂彩券。假設每張彩券贏得某個獎項的機會是 1/9 ，而彩券彼此互相獨立。因此每張彩券可視為一次 Bernoulli trial ；若令 X 代表會中獎的彩券張數，則 X 具有 n = 10, p = 1/9 的binomial distribution 。

• 則

• 老王的彩券至少有三張會中獎的機率，便是

.10,,2,1,0 ,9

8

9

110)(

10

xx

xpxx

X

0906.09

8

9

110)()3(

1010

3

10

3

xx

xxX x

xpXP


Means and variances for binomial random variables

np

qpnp

qpx

nnp

qpx

nxX

n

n

x

xnx

n

x

xnxX

)(

1

1

][E

1

1

)1()1(1

0


Means and variances for binomial random variables (contd.)

•

•

)1()]1([E)]1([E

have we

)]1([E][E)]1([E][E Since

22

2

XXXXX

X

XXXX

XXXXXX

2

22

2

22

20

)1(

)()1(

)!()!2(

)!2()1(

)!(!

!)1()1()]1([E

pnn

qppnn

qpxnx

npnn

qpxnx

nxxqp

x

nxxXX

n

n

x

xnx

n

x

xnxn

x

xnx


Means and variances for binomial random variables (contd.)

• Thus

npq

pnp

npnppnn

XX XXX

)1(

)1()1(

)1()]1([E2

2


• Before introducing the other probability distribution, we have to be familiar to infinite geometric series first.


Infinite geometric series

• When | q | < 1,

1

1

1

1 limlim

1

0 qq

qq

n

n

n

i

i

n

)1(

1

1

12

0 qqdq

dq

dq

d

i

i

)1(

2

)1(

132

0

2

qqdq

dq

dq

d

i

i


Infinite geometric series (contd.)

• Then we will obtain that

• An exercise.

10 )1(

1

k

kj

kj

i

i

qq

k

jq

i

ik


Geometric distribution ( 幾何分佈 )

• Let N be the trial number of the first successthe first success in a sequence of independent Bernoulli trials, each with parameter p. The probability function for N is

N is called a geometric random variable with parameter p.

otherwise. ,0

},3,2,1{for ,)(

1 Nn

N

Rnpqnp


Memoryless property ( 失憶性 )

• If N is a geometric random variable with parameter p, then

where a and b are any positive integers. This is the only discrete probability law to have this memoryless property.

).()|( aNPbNbaNP


• 舉例來說：

• 假設我們現在要搜尋一個得 SARS 的病患，而當我們找到第一個病患就停止搜尋。不同的人之間為互相獨立的 Bernoulli trials ， p = 0.1 。

• 假設我們已經檢查了 8 個人，都還沒出現成功的試驗 ( 找到一個得SARS 的病患 ) ，則下一個人是 SARS 病患的機率並不會因此改變。這即為失憶性 (memoryless property) 。


Means and variances for geometric random variables

•

p

qp

qqp

pqnnpnNn

n

nNN

1

)1(

1

)321(

)(][E

2

2

1

1

1


Means and variances for geometric random variables (contd.)

• Since

We have

23

2

2

1

1

2

)1(

2

)1262(

)1(

)()1()]1([E

p

q

qpq

qqpq

pqnn

npnnNN

n

n

nN

22 1

112

)1()]1([E][Var

p

q

ppp

q

NNN NN


Negative binomial distribution ( 負二項分佈 )

• Independent Bernoulli trials, each with probability of success p, are performed until the rth success occurs. The number of trials required, Nr , is called a negative binomial random variable with parameter r, p; its probability function is as follows:

otherwise. ,0

},2,1,{ ,1

1)(

rrrRnqpr

nnp N

rnr

Nr


Means and variances for negative binomial random variables

•

2

,

p

rq

p

r

r

r

N

N


Means and variances for negative binomial random variables (contd.)

•

•

p

r

qrpq

r

nnpN

rnr

rrnrr

1)1(

1

1

1][E

2

1,1 where,2)1(

2)!(!

)!1(

)!(!

!)21(

)!()!1(

)!1()1()]1([E

p

qprr

rrnnqr

nrpq

r

nprr

qr

nrpq

rnr

nrp

qrnr

nnrp

qprnr

nnnNN

rn

rnr

rn

rnr

rn

rnr

rn

rnr

rn

rnr

rn

rnrrr


Means and variances for negative binomial random variables (contd.)

• Thus

2

22

1

p

rq

p

r

p

r

p

qprr

rN

Thank you.


References

• [H01] 黃文典教授 , 機率導論講義 , 成大數學系 , 2001.

• [L94] H. J. Larson, Introduction to Probability, Addison-Wesley Advanced Series in Statistics, 1994; 機率學的世界，鄭惟厚譯，天下文化出版。

• [M97] Statistics: Concepts and Controversies, David S. Moore, 1997; 統計，讓數字說話，鄭惟厚譯 , 天下文化出版。

• [MR95] R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press, 1995.

Documents

Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation