Transcript Lecture 14

UNR, MATH/STAT 352, Spring 2007
Binomial(n,p)
n k
n!
n k
P( X  k )    p (1  p ) 
p k (1  p ) n k
(n  k )! k !
k 
E ( X )  np
  np(1  p )
2
X
UNR, MATH/STAT 352, Spring 2007
0.1
0.2
0.3
Number of successes within 1
symmetric Bernoulli trial
can only be 0 or 1. These
possibilities have equal chances.
0.0
dbinom(q, 1, 0.5)
0.4
0.5
Binomial(1,0.5)
0
2
4
6
Number of succsses
UNR, MATH/STAT 352, Spring 2007
8
10
0.5
Binomial(2,0.5)
0.3
0.2
0.1
dbinom(q, 2, 0.5)
0.4
Number of successes within 2
symmetric Bernoulli trials
can only be 0, 1 or 2.
The possibility to have exactly 1 success
is larger than that of having 0 or 2.
0.0
A fair game should result in a tie.
0
2
4
6
8
10
Then
why
do people play fair games?
Number of
succsses
UNR, MATH/STAT 352, Spring 2007
0.8
Binomial(2,0.1)
0.4
0.2
0.0
dbinom(q, 2, 0.1)
0.6
Most likely there will be no successes
0
2
4
6
Number of succsses
UNR, MATH/STAT 352, Spring 2007
8
10
0.8
Binomial(2,0.9)
0.4
0.2
0.0
dbinom(q, 2, 0.9)
0.6
Most likely there will be only successes
0
2
4
6
Number of succsses
UNR, MATH/STAT 352, Spring 2007
8
10
0.20 0.25
Unimodal (mode = 1)
Right-skewed (E = np = 1.5)
0.05 0.10
0.15
Concentrated around 1.5
0.00
dbinom(q, 15, 0.1)
0.30
0.35
Binomial(15,0.1)
0
2
4
6
Number of succsses
UNR, MATH/STAT 352, Spring 2007
8
10
Unimodal (mode = 1-2)
0.20
Right-skewed
0.05
0.10
0.15
Concentrated around 1.5
0.00
dbinom(q, 19, 0.1)
0.25
Binomial(19,0.1)
0
2
4
6
Number of succsses
UNR, MATH/STAT 352, Spring 2007
8
10
0.12
Unimodal (mode = 10)
0.10
Symmetric? (E = np = 10)
0.08
Concentrated around 10
0.02
0.04
0.06
P(10+3) < P(10-3)
0.00
dbinom(q, 100, 0.1)
Binomial(100,0.1)
0
5
10
15
20
Number of succsses
UNR, MATH/STAT 352, Spring 2007
25
30
0.12
0.10
Symmetric? (E = np = 90)
Concentrated around 90
0.02
0.04
0.06
0.08
Unimodal (mode = 90)
0.00
dbinom(q, 100, 0.9)
Binomial(100,0.9)
70
75
80
85
90
Number of succsses
UNR, MATH/STAT 352, Spring 2007
95
100
Bin(100,0.9)
0.04
0.06
0.08
0.10
Only a small fraction of possible
outcomes has not negligible P
(i.e. only small part can be seen
in experiment)
0.02
P is very small (not 0!) here
0.00
dbinom(q, 100, 0.1)
0.12
Bin(100,0.1)
0
20
40
60
Number of succsses
UNR, MATH/STAT 352, Spring 2007
80
100
0.30
Binomial(6,0.5)
0.20
0.15
0.10
0.05
0.00
dbinom(q, 6, 0.5)
0.25
Here all possible
outcomes have reasonable
probabilities
0
1
2
3
4
Number of succsses
UNR, MATH/STAT 352, Spring 2007
5
6
Poisson(l)
P( X  k )  e  l
E( X )  l
lk
k!
 l
2
X
UNR, MATH/STAT 352, Spring 2007
Poisson(1)
0.3
0.2
0.1
0.0
0.2
0
2
4
6
8
0.1
Number of succsses
0.0
dpois(q, 1)
0.3
dbinom(q, 1000, 1/1000)
Binomial(1000,.001]
0
2
4
6
Number of succsses
8
I’ve seen this already!
UNR, MATH/STAT 352, Spring 2007
10
10
n 
+
p  0 + np  l
Binomial(n,p)  Poisson(l)
UNR, MATH/STAT 352, Spring 2007
Poisson(30)
Binomial(1000,.03) N(30,sqrt(30))
0.06
Poisson
0.02
0.04
Normal
0.00
dpois(q, 30)
Binomial
0
10
20
30
Number of succsses
UNR, MATH/STAT 352, Spring 2007
40
50
If n is large (n > 100), p is small (p < 0.05), and
both np and n(1-p) are not small (say >10) then
B(n,p)~P(np)~N(np, np(1-p))
UNR, MATH/STAT 352, Spring 2007
0.7
Normal densities
0.5
0.6
Mean= 0, Std=0.6
0.4
0.3
Mean= 0, Std=1
0.1
0.2
Mean= 5, Std=1
Mean= 0, Std=3
0.0
Density
Mean= 7, Std=1
-10
-8
-6
-4
-2
0
2
4
Value of random variable
UNR, MATH/STAT 352, Spring 2007
6
8
10
0.006
Interval probability
Lower-tail probability
Upper-tail probability
0.000
Probability
0.012
Probability density function
2400
2450
2500
2550
2600
Value of random variable
0.8
0.4
0.0
Probability
Cumulative distribution function
2400
2450
2500
2550
Value of random variable
UNR, MATH/STAT 352, Spring 2007
2600
UNR, MATH/STAT 352, Spring 2007
Payoff  2 X  N , where X Bin(N , 1 2 )
(Why this formula? See lecture notes.)
0.3
0.2
0.1
0.0
Probability of payoff
0.4
0.5
2 games. P(lose or win < $10)= 1
-100
-50
0
50
Net payoff, in $
UNR, MATH/STAT 352, Spring 2007
100
Payoff  2 X  N , where X Bin(N , 1 2 )
(Why this formula? See lecture notes.)
0.3
0.2
0.1
0.0
Probability of payoff
0.4
0.5
4 games. P(lose or win < $10)= 1
-100
-50
0
50
Net payoff, in $
UNR, MATH/STAT 352, Spring 2007
100
Payoff  2 X  N , where X Bin(N , 1 2 )
(Why this formula? See lecture notes.)
0.3
0.2
0.1
0.0
Probability of payoff
0.4
0.5
6 games. P(lose or win < $10)= 1
-100
-50
0
50
Net payoff, in $
UNR, MATH/STAT 352, Spring 2007
100
Payoff  2 X  N , where X Bin(N , 1 2 )
(Why this formula? See lecture notes.)
0.3
0.2
0.1
0.0
Probability of payoff
0.4
0.5
10 games. P(lose or win < $10)= 1
-100
-50
0
50
Net payoff, in $
UNR, MATH/STAT 352, Spring 2007
100
Payoff  2 X  N , where X Bin(N , 1 2 )
(Why this formula? See lecture notes.)
0.3
0.2
0.1
0.0
Probability of payoff
0.4
0.5
100 games. P(lose or win < $10)= 0.728
-100
-50
0
50
Net payoff, in $
UNR, MATH/STAT 352, Spring 2007
100
Payoff  2 X  N , where X Bin(N , 1 2 )
(Why this formula? See lecture notes.)
0.3
0.2
0.1
0.0
Probability of payoff
0.4
0.5
1000 games. P(lose or win < $10)= 0.272
-100
-50
0
50
Net payoff, in $
UNR, MATH/STAT 352, Spring 2007
100
Payoff  2 X  N , where X Bin(N , 1 2 )
(Why this formula? See lecture notes.)
0.3
0.2
0.1
0.0
Probability of payoff
0.4
0.5
10000 games. P(lose or win < $10)= 0.087
-100
-50
0
50
Net payoff, in $
UNR, MATH/STAT 352, Spring 2007
100