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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
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 33
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.
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 44
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
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 55
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
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 66
• 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.
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 77
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
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 88
• 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
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 99
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
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1010
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
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1111
Means and variances for binomial random variables (contd.)
• Thus
npq
pnp
npnppnn
XX XXX
)1(
)1()1(
)1()]1([E2
2
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1212
• Before introducing the other probability distribution, we have to be familiar to infinite geometric series first.
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1313
Infinite geometric series
• When | q | < 1,
1
1
1
1 limlim
1
0 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
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1414
Infinite geometric series (contd.)
• Then we will obtain that
• An exercise.
10 )1(
1
k
kj
kj
i
i
k
jq
i
ik
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1515
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
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1616
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
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1717
• 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) 。
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1818
Means and variances for geometric random variables
•
p
qp
qqp
pqnnpnNn
n
nNN
1
)1(
1
)321(
)(][E
2
2
1
1
1
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1919
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
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 2020
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
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 2121
Means and variances for negative binomial random variables
•
2
,
p
rq
p
r
r
r
N
N
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 2222
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
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 2323
Means and variances for negative binomial random variables (contd.)
• Thus
2
22
1
p
rq
p
r
p
r
p
qprr
rN
Thank you.
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 2525
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.