Download Bernoulli random variable with parameter p

Introduction to Probability Theory ‧31‧
- Preliminaries for Randomized Algorithms
Speaker: Chuang-Chieh Lin
Advisor: Professor Maw-Shang Chang
National Chung Cheng University
Dept. CSIE, Computation Theory Laboratory
January 25, 2006
Outline
• Chapter 3: Discrete random variables
– Bernoulli and binomial distributions
– Geometric distribution
– Negative binomial distribution
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Bernoulli trials (伯努利試驗)
• A Bernoulli trial is an experiment with two different
possible outcomes, labeled success and failure. 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.
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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:
 p x q1 x , for x  0, 1
p X ( x)  
otherwise
0,
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Bernoulli random variable (contd.)
• Mean and variance for a Bernoulli random variable X
with parameter p:
E[ X ]  0  p X (0)  1 p X (1)  p  E[ X 2 ]
 X2  E[ X 2 ]  (E[ X ]) 2  p  p 2  p(1  p)  pq
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• 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.
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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
 n  y n  y
  p q ,
pY ( y )   y 
 0,

for y  RY .
otherwise.
where q = 1 – p
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• 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
10   1 
p X ( x)     
 x  9 
x
10 x
8
 
9
, x  0,1, 2,,10.
• 老王的彩券至少有三張會中獎的機率，便是
10   1 
P( X  3)   p X ( x)      
x 3
x 3  x   9 
10
10
x
10 x
8
 
9
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 0.0906
8
Means and variances for binomial random
variables
 n  x n x
 X  E[ X ]   x  p q
x 0  x 
n
 n  1 x 1 ( n 1) ( x 1)
 p q
 np  
x 1  x  1
 np ( p  q ) n 1
n
 np
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Means and variances for binomial random
variables (contd.)
• Since E[ X 2 ]  E[ X ( X  1)]  E[ X ]  E[ X ( X  1)]   X
we have
 X2  E[ X ( X  1)]   X   X2  E[ X ( X  1)]   X (  X  1)
•
n
 n  x n x
n!
E[ X ( X  1)]   x( x  1)  p q
  x( x  1)
p x q n x
x!( n  x)!
x 0
x2
 x
n
(n  2)!
2
 n(n  1) p 
p x2 q n x
x  2 ( x  2)! ( n  x )!
n
 n(n  1) p 2 ( p  q ) n  2
 n(n  1) p 2
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Means and variances for binomial random
variables (contd.)
• Thus
 X2  E[ X ( X  1)]   X (  X  1)
 n(n  1) p 2  np(np  1)
 np(1  p)
 npq
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• Before introducing the other probability distribution,
we have to be familiar to infinite geometric series
first.
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Infinite geometric series
• When | q | < 1,
n
lim
n
i
q
  lim
i 0
n
1  q n1
1

1 q
1 q
d   i d  1

q  
dq  i 0  dq  1  q
d2   i  d 
1

q  
dq  i 0  dq  (1  q) 2

1
 
2
(
1

q
)


2
 
3
(
1

q
)

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Infinite geometric series (contd.)
• Then we will obtain that
 k  i  i   j  j k
1

 q    q 

k 1
i
k
(
1

q
)
i 0 
j k  


• An exercise.
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Geometric distribution (幾何分佈)
• Let N be the trial number of the first success in a
sequence of independent Bernoulli trials, each with
parameter p. The probability function for N is
 pq n 1 ,
p N ( n)  
 0,
for n  RN  {1, 2, 3,}
otherwise.
N is called a geometric random variable with
parameter p.
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Memoryless property (失憶性)
• If N is a geometric random variable with parameter p,
then
P( N  a  b | N  b)  P( N  a).
where a and b are any positive integers. This is the
only discrete probability law to have this memoryless
property.
Computation Theory Lab., Dept. CSIE, CCU, Taiwan
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• 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)。
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Means and variances for geometric random
variables
•


n 1
n 1
 N  E[ N ]   n  p N (n)   n  pq
n 1
 p (1  2q  3q 2  )
1
p
(1  q ) 2
1

p
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Means and variances for geometric random
variables (contd.)

• Since E[ N ( N  1)]   n(n  1) p N (n)
n 1

  n( n  1) pq n 1
n2
 pq ( 2  6q  12q 2  )
2
2q
 pq
 2
3
(1  q )
p
We have Var[ N ]  E[ N ( N  1)]   N (  N  1)

2q 1  1
q
 2    1  2
p
p p
 p
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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:
 n  1 r n  r
 p q , n  RN  {r , r  1, r  2,}

p N r (n)   r  1 
 0,
otherwise.

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Means and variances for negative binomial
random variables
•
N
r
N
r
r
 ,
p
rq
 2
p
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Means and variances for negative binomial
random variables (contd.)
 n  1 n r
1
r
r
 q  rp
• E[ N r ]   np 

r 1
(1  q)
p
nr
 r  1

r
•

E[ N r ( N r  1)]   n(n  1)
nr
 rp

r
 (n  1  2)
nr
(n  1)!
p r q nr
(r  1)!(n  r )!
n!
q nr
r!(n  r )!

n
(n  1)! n  r
r
 rp 
q  2rp   q n  r
n  r r!( n  r )!
nr  r 

r

 n  n r 
n
r
 r (r  1) p   q
 2rp   q n  r , where n  n  1, r   r  1
n  r   r  
nr  r 
r pq
 r
p2

r
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Means and variances for negative binomial
random variables (contd.)
• Thus

2
Nr

r pq r  r
 r
   1
2
p
p p

rq
 2
p
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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.
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