Bài giảng Probability & Statistics - Lecture 5: Discrete probability - Bùi Dương Hải
Example Example 5.6. The quiz includes 10 multiple choices questions, each has 4 options and only one correct. A candidate do all questions by random choose the answers. (a) Expectation and variance of number of correct answer? (b) Probability that there are 3 correct answers? (c) Probability that there are at least 6 correct ones? (d) Each correct one is evaluated (+4) points, but for incorrect one, it is (-1) point. What is the chance for candidate gain 10 points in total ?
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Lecture 5. DISCRETE PROBABILITY
Random Variable
Probability Distribution
Expected value
Variance – Standard Deviation
Bivariate Probability
Binomial Distribution
[1] Chapter 5: pp.215 - 260
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5.1. Random Variable
Random variable: numerical value from a random
experiment.
Denoted by X, Y, Z, or X1, X2,...
Ex. Tossing a die, X is the number of dots
- Number of boys in a 3-children family
- Score of students’ exam
- Temparature during a day
- Interest rates in a period of time
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Types of Random Variable
Variable , value is random
Discrete variable: = ( , , , )
Number of item: = (0, 1, 2, )
Score of test: = (0, 1, 2, , 100)
( = ) is a random event
Continuous variable: = ( ; )
Time
Temperature
Length, Weight
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5.2. Discrete Probability Distribution
Discrete: = ( , , , )
Denote: = =
Property: ∑
= 1
is discrete probabitily distribution; probabitiy
function
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Value
Probability
Example
Ex. Probability distribution of X, which is the number of
Heads when flipping a coin twice
X = {0, 1, 2}
Example 5.1. Number of Head when flipping a coin 3
times
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x 0 1 2
P(x) 1/4 2/4 1/4
Flip a coin twicep
X 0 1 2 3
Probability
5.3. Parameter
Parameter of Random variable:
Expected value (Mean)
Variance, Standard Deviation
Ex.
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Salary ($) 7 8 9
Frequency 2 5 3
Percent 20% 50% 30%
Probability 0.2 0.5 0.3
Expected Value
Expected value of X, denoted by E(X) or μX
= = ∑
Expected value of X is also Population Mean, and has
the same unit with X.
Properties: if is a constant
( ) =
( + ) = ( ) +
( ) = ( )
( ± ) = ( ) ± ( )
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Variance – Standard Deviation
Variance of is denoted by ( ) or ( ) or
= −
= ∑ −
Unit of Variance is square of unit of
Standard Deviation of is denoted by ( ) or
= ( )
Unit of Variance is unit of
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Comparison
Example 5.1. Compare return rate of three projects
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Project
B
Return rate (%) 5 15
Probability 0.5 0.5
Project
C
Return rate (%) –10 10 24
Probability 0.2 0.3 0.5
Project
A
Return rate (%) 7
Probability 1
Properties of E(X) and V(X)
, are variable; is constant
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Expected Value Variance
= = 0
+ = + + =
× = × × = ×
± = ± ( ) ± =
+ ± 2 ( , )
± = +
If X and Y are independent
Investment
Example 5.3. There are 4 independent projects, each
have the same return rate probability distribution:
Expected value and Variance when:
(a) Invest 10 ($ mil.) in one project
(b) Invest 40 ($ mil.) in one project
(c) Invest in 4 projects, each 10 ($ mil.)
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Return rate (%) 0 20
Probability 0.3 0.7
5.4. Bivariate Probability
Example 5.5. Profit of Project 1 and 2 are and ,
respectively, with Bivariate Probability table:
Fill the blanks
, , , , + , ( + )?
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X Y – 1 0 5
–2 0.05 0.1 0.05 0.2
7 0.05 0.2 0.55 0.8
0.1 0.3 0.6 1
Covariance and Correlation
Covariance
, = −
= ∑ ∑ . . − ( )
± = + ± 2 ( , )
± = + ± 2 ( , )
Correlation
, =
,
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Porfolio
Ex. Two investment projects A and B
= −6
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Project A B
E(return) 10 20
(return) 5 12
% for A % for B ( ) ( )
100% 0% 10 5.00
90% 10% 11 4.54
80% 20% 12 4.45
70% 30% 13 4.76
60% 40% 14 5.40
50% 50% 15 6.26
40% 60% 16 7.28
30% 70% 17 8.38
20% 80% 18 9.55
5.5. Binomial Distribution
Bernoulli problem: independent experiments,
probability of .
is number of success
Distribution of X is Binomial: ~ ( , )
= , =
−
=
= − ; = ( − )
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Binomial Distribution
Binomial Table (Table)
Ex.
( = 1| = 3, = 0.2) = 0.384
( = 6| = 10, = 0.3) = 0.1029
( = 4| = 10, = 0.7) = 0.1029
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n x P
.20
3 0 .5120
1 .3840
2 .0960
3 .0080
Example
Example 5.6. The quiz includes 10 multiple choices
questions, each has 4 options and only one correct. A
candidate do all questions by random choose the
answers.
(a) Expectation and variance of number of correct
answer?
(b) Probability that there are 3 correct answers?
(c) Probability that there are at least 6 correct ones?
(d) Each correct one is evaluated (+4) points, but for
incorrect one, it is (-1) point. What is the chance for
candidate gain 10 points in total ?
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5.6. Poisson Distribution
Denoted: ~ ( )
=
!
= 0,1,2
= ; =
Binomial Distribution with large and small (that
(1 – )) converges to Poisson Distribution,
with l = .
Ex. The number of mistake papers of a photo machine in
one day is Poisson distribution with mean of 3.
Find the probability that in the following day, there will
be 4 mistake papers
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Poisson Distribution – Table
Ex. ~ ( = 3)
= 4 = 3 =
!
=
Using Table 7 (p.995), =
= 4 = 3 =
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x
3.0
0 .0498
1 .1494
2 .2240
3 .2240
4 .1680
5 .1008
6 .0504
Example 5.7. The probability that a passenger forgets his
(her) luggage on train is 0.008. What the probability that
in 400 passengers, there is
(a) No forgotten luggage
(b) At least 4 forgotten luggages
Key Concepts
Random Variable
Discrete Variable
Probability Distribution
Expected Value
Variance, Standard Deviation
Bivariate Probability Distribution
Covariance
Binomial Distribution, Poisson Distribution
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Exercise
[1] Chapter 5:
(227) 16, 20, 21, 22
(237) 25, 26, 28
(248) 32, 35, 38,
(260) 60, 66, 67,
[1] Case Study : Hamilton County Judges
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