Lecture Probability & statistics: Chapter 5

Lecture 5. DISCRETE PROBABILITY Lecture 5. DISCRETE PROBABILITY

 Random Variable  Probability Distribution  Expected value  Variance – Standard Deviation  Bivariate Probability  Binomial Distribution

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 [1] Chapter 5: pp.215 - 260

5.1. Random Variable 5.1. Random Variable

 Random variable: numerical value from a random

experiment.

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 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

Types of Random Variable 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: = (; )

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 Time  Temperature  Length, Weight

5.2. Discrete Probability Distribution 5.2. Discrete Probability Distribution

 Discrete: = (, , … , )  Denote: = =

= 1

… … Value Probability

 Property: ∑  is discrete probabitily distribution; probabitiy

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function

Example Example

Flip a coin twice

x P(x)

1/4

2/4

1/4

Ex. Probability distribution of X, which is the number of Heads when flipping a coin twice X = {0, 1, 2} p

Example 5.1. Number of Head when flipping a coin 3 times

0 1 2 3

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X Probability

5.3. Parameter 5.3. Parameter

 Parameter of Random variable:  Expected value (Mean)  Variance, Standard Deviation  Ex.

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Salary ($) Frequency Percent Probability 7 2 20% 0.2 8 5 50% 0.5 9 3 30% 0.3

Expected Value 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 – Standard Deviation

 Variance of is denoted by () or () or

= − = ∑ −

Unit of Variance is square of unit of

 Standard Deviation of is denoted by () or

= ()

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Unit of Variance is unit of

Comparison Comparison

Example 5.1. Compare return rate of three projects

Project A Return rate (%) Probability 7 1

Project B Return rate (%) Probability 5 0.5 15 0.5

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Project C Return rate (%) Probability –10 0.2 10 0.3 24 0.5

Properties of E(X) and V(X) Properties of E(X) and V(X)

 , are variable; is constant

Expected Value = Variance = 0

+ = +

× = × + = × = ×

± = ± () ± =

+ ± 2(, )

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± = + If X and Y are independent

Investment Investment

Example 5.3. There are 4 independent projects, each have the same return rate probability distribution:

Return rate (%) Probability 0 0.3 20 0.7

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 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.)

5.4. Bivariate Probability 5.4. Bivariate Probability

Example 5.5. Profit of Project 1 and 2 are and , respectively, with Bivariate Probability table:

 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

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 Fill the blanks  , , , , + , ( + )?

Covariance and Correlation Covariance and Correlation

 Covariance

, = − = ∑ ∑ . . − ()  ± = + ± 2(, )  ± = + ± 2(, )

 Correlation

, =

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Porfolio Porfolio

 Ex. Two investment projects A and B

A 10 5 B 20 12 Project E(return) (return)

 = −6

100% 90% 80% 70% 60% 50% 40% 30% 20%

0% 10% 20% 30% 40% 50% 60% 70% 80%

10 11 12 13 14 15 16 17 18

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% for A % for B () () 5.00 4.54 4.45 4.76 5.40 6.26 7.28 8.38 9.55

5.5. Binomial Distribution 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 Distribution

n

x

0 1 2 3

Binomial Table (Table)

P … .20 … .5120 .3840 .0960 .0080

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Ex. ( = 1| = 3, = 0.2) = 0.384 ( = 6| = 10, = 0.3) = 0.1029 ( = 4| = 10, = 0.7) = 0.1029

Example 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

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incorrect one, it is (-1) point. What is the chance for candidate gain 10 points in total ?

5.6. Poisson Distribution 5.6. Poisson Distribution

 Denoted: ~()

 = = 0,1,2 …

 = ; =  Binomial Distribution with large and small (that (cid:0) (1 – )) converges to Poisson Distribution, with l = .

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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

Poisson Distribution – Table Poisson Distribution – Table

x

Ex. ~( = 3)

= 4 = 3 =

=

Using Table 7 (p.995), = = 4 = 3 =

0 1 2 3 4 5 6 …

… 3.0 … .0498 .1494 .2240 .2240 .1680 .1008 .0504 …

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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 Key Concepts

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 Random Variable  Discrete Variable  Probability Distribution  Expected Value  Variance, Standard Deviation  Bivariate Probability Distribution  Covariance  Binomial Distribution, Poisson Distribution