Lecture "Probability & statistics - Chapter 6: Continuous probability" has contents: Continuous random variable, density function, parameter, uniform distribution, normal distribution, cutoff point. Invite you to consult the content.
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Nội dung Text: Lecture Probability & statistics: Chapter 6 - Bùi Dương Hải
- Lecture 6. CONTINUOUS PROBABILITY
Continuous Random Variable
Density Function
Parameter
Uniform Distribution
Normal Distribution
Cutoff point
[1] Chapter 6. pp. 255 - 294
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 1
- 6.1. Continuous Random Variable
Continuous Random Variable: uncountable values
Available value is one interval: = ( , )
Maybe: = −∞; = +∞
Probability that one point: = =0
Consider Probability at one interval: ( < < )
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 2
- 6.2. Density Function
Discrete Continuous
X … X ( , )
Prob. … Density ( )
∑ =1 ∫ =1
p f(x)
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 3
- Density Function
≥0
∫ =1
< < =∫
Cutoff point level denoted by : > =
f(x)
a b
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 4
- 6.3. Parameter
Expected Value:
= =∫
Variance:
=∫ −
=∫ −
Standard Deviation
= ( )
Cutoff point level , denoted by :
> =
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 5
- Example
Example 6.1. Waiting time (hour), with density
function
2 ∈ [0,1]
=
0 ∉ [0,1]
(a) Prob. of waiting more than a half of hour?
(b) Prob. of waiting from 20 to 40 minutes?
(c) The average and variance of waiting time?
(d) Cutoff point level 10%?
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 6
- Example
. f(x)
(a) > 0.5 = ∫ 2 .
/
(b) < < = ∫/ 2 .
0.5
f(x)
(c) =∫ .2 .
=∫ .2 . −
1/3 2/3
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 7
- 6.4. Uniform Distribution
~ ( , ) if
∈[ , ]
=
0 ∉[ , ]
a c d b
= ; =
< < =
Ex. Temperature is Uniform Distribution in the interval of
(20, 30)oC. What is the probability that temperature is
between 23 and 28 degree?
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 8
- 6.5. Normal Distribution
, = 0.5 : = 10; 20; 100 Normality
0.25 0.25
0.2 0.2
0.15 0.15
0.1 0.1
0.05 0.05
0 0
1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
0.025 0.025
0.02 0.02
0.015 0.015
0.01 0.01
0.005 0.005
0 0
1 6 111621263136414651566166717681869196 0 20 40 60 80 100
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 9
- Normal Distribution
Density Function: =
Denoted: ~ ( , )
=
=
=
1
σ 2π
f(x)
μ μ’
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- Normal Distribution
Carl Friedrich Gauss
(1777-1855) in 1809
0.6
~ 3,1 0.5
~ 6,1 0.4
0.3
~ (8,0.5 ) 0.2
0.1
0
0 1 2 3 4 5 6 7 8 9 10
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- Standardized Normal Variable
0.5
~ , 0.4
= 0.3
0.2
~ (0,1) 0.1
0
-4 -3 -2 -1 0 1 2 3 4
Table 1
< 1 = 0.8413
< 1.25 =
>2 =
−1 < < 1.3 =
0.5
1.5
2.5
3.5
-4
-3
-2
-1
-3.5
-2.5
-1.5
-0.5
0
1
2
3
4
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 12
- Probability formula
~ ,
− − −
< = < = <
Ex. ~ 100,16
< 104 =
> 92 =
94 < < 102 =
Probability that X differ from the mean not more than
standard deviation =
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 13
- Example
Example 6.2. Return ($mil) of project A is normality with
mean of 8 and variance of 9. Calculate the probability:
(a) Return of A higher than 10
(b) Loss money
(c) Return of A between 5 and 12
Return of project B is normality with mean of 10 and
variance of 25. A and B are independent. Calculate the
probability that:
(c) Both gain positive return
(d) Total return of A and B greater than 20
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 14
- 3-sigma Rule
− < < + = 68.26%
−2 < < +2 = 95.44%
−3 < < +3 = 99.75%
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 15
- Cutoff point
Cutoff point level , or “critical value”
Denoted:
> =
> 1.96 = 0.025 . = 1.96
> 1. 64 = 0.0505 . = 1.64
> 1. 65 = 0.0495 . = 1.65
Keys: . = . ; . = .
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 16
- 6.6. Binomial vs Normal
Binomial: ~ ( , ) with ≥ 100
approximate: ( , )
With: = ; = (1 − )
Example 6.3. Probability that visitor buy good in the
shopping mall is 0.3. In 400 visitors, what is the
probability
(a) There are at least 100 buyers
(b) Number of buyers is from 90 to 150
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 17
- 6.7. Cutoff Point
Normal Distribution:
Student Distribution:
df: Degree of freedom
Table 2 (p.976)
. = 1.833; . = 2.086
≈
Chi-square Distribution:
Table 3 (p.979)
. = 3.94 ; . = 24.996
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 18
- Key Concepts
Continuous variable
Density function
Normal distribution
Exercise
[1] Chapter 6:
(270) 3, 5
(281) 11, 12, 17, 19, 23, 24, 31
(292) 41, 44, 49
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