Lesson 1 Chapter 5 probability

Statistics 2Statistics 2

Dr. Ning DING

[email protected]

I.007, IBS

Lesson 01, Sept 2011

Table of Contents

1. What is Probability Distribution?

2. Random Variables

3. Use of Expected Value in Decision Making

4. The Binomial Distribution

5. The Poisson Distribution

6. The Normal Distribution: A Distribution of a Continuous Random Variable

Probability Distribution?

Random Variables

Expected Value in Decision Making

The Binomial Distribution

The Poisson Distribution

The Normal Distribution

What is a probability distribution?

= theoretical frequency distribution

AnimationDemo

= listing of the probabilities of all the possible outcomes


Random Variables






Random Variables







Random Variables






• Discrete

• Continuous

= only a limited number of values

= any value within a given range, e.g. height or weight


Random Variables





Types of probability distribution?

• Can be either discrete or continuous

Chapter 5, SC 5-2 P.229


Random Variables





Random Variables

• What did you expect?


Average number of women screened daily is: 108.02


Random Variables





The Expected Value of Random Variables

• Combine probabilities and monetary values



Random Variables





Use of Expected Value in Decision Making

= probability distribution of a discrete random variable

Probability of r successes in n trials =n!

r ! (n-r) !

pr qn-r

Number of trials

Number of successes desired

Probability of success

Probability of failure

factorial!

阶乘 Fakultät factorial la factorielle


Random Variables







• Use of the Bernoulli Process


r ! (n-r) !

pr qn-r

2 tails in 3 tosses?

Probability of 2 successes in 3 trials =3!

2 ! (3-2) !0.52 *0.53-2

3*2*1

2 *1 *1*1 !0.52 *0.5

=0.3750

Example:Example:Probability Distribution?

Random Variables







Random Variables





= for discrete random variables

Mean of successses

Probability of x occurences

2.71828

x factorial



Random Variables







Random Variables





Example:Example:Mean=5 accidents per month;

Calculate the probability of exactly 0,1,2,3,or 4 accidents

Step 1: look up in Table 4a to get e e = 0.00674

Step 2: Calculate the P

P=0.12469P=0.26511

Using Table to solve the problem

Table 4 (b)Table 4 (b)



Random Variables





Characteristics of the Normal Probability Distribution

UnimodalUnimodal Mean at the center

Mean at the center

Mean=median=mode

Mean=median=modeTails off indefinitely

Tails off indefinitely


Random Variables






The same meanDifferent SD

The same meanDifferent SDProbability

Distribution?

Random Variables






The same SDDifferent meanThe same SD

Different meanProbability Distribution?

Random Variables






The largest SD?The largest mean?

The largest SD?The largest mean?


Random Variables







Random Variables






z score

Mean length of time is 500 hours;SD is 100 hours;

Q1. What is the probability of >500 hours?

Q2. What is the probability of 500~650 hours?


Random Variables






0.500.50

0.43320.4332

z score


Q3. What is the probability of >700 hours?



Random Variables






0.5-0.4772=0.02280.5-0.4772=0.0228

0.4332-0.1915 = 0.24170.4332-0.1915 = 0.2417

z score


Q5. What is the probability of <580 hours?



Random Variables






0.2881+0.5=0.78810.2881+0.5=0.7881

0.2580+0.2881=0.54610.2580+0.2881=0.5461


Random Variables






z score

Mean of the distributionMean of the distribution

SD of the distributionSD of the distribution

valuevalue

Summary

1. What is Probability Distribution?

2. Random Variables

3. Use of Expected Value in Decision Making

4. The Binomial Distribution

5. The Poisson Distribution

6. The Normal Distribution: A Distribution of a Continuous Random Variable


Random Variables






Many utility companies promote energy conservation by offering discount rates to consumers who keep their energy usage below certain established subsidy standards. A recent EPA report notes that 70% of the island residents of Puerto Rico have reduced their electricity usage sufficiently to qualify for discounted rates. If ten residential subscribers are randomly selected from San Juan, Puerto Rico, what is the probability that at least four qualify for the favorable rates?

Test yourselfTest yourself

Extra Reading

Extra Reading



Random Variables






r ! (n-r) !

pr qn-r

Example: Pat Statsdud is a student taking a statistics course. Unfortunately, Pat is not a good student. Pat does not read the textbook before class, does not do homework, and regularly misses class. Pat intends to rely on luck to pass the next quiz. The quiz consists of 10 multiple-choice questions. Each question has five possible answers, only one of which is correct. Pat plans to guess the answer to each question.

a. What is the probability that Pat gets no answers correct?


0 ! (10-0) !

0.20 0.810-0

P(0)= .1074

Example:Example:

Extra Reading

Extra Reading


r ! (n-r) !

pr qn-r

Example: Pat Statsdud is a student taking a statistics course. Unfortunately, Pat is not a good student. Pat does not read the textbook before class, does not do homework, and regularly misses class. Pat intends to rely on luck to pass the next quiz. The quiz consists of 10 multiple-choice questions. Each question has five possible answers, only one of which is correct. Pat plans to guess the answer to each question.

b. What is the probability that Pat gets two answers correct?


2 ! (10-2) !

0.22 0.810-2

P(2)= .3020



Random Variables





Example:Example:

Extra Reading

Extra Reading


$20 per case

$30 per case


Random Variables





Combining Probabilities and Monetary Values Example:Example:

Extra Reading

Extra Reading

– Obsolescence Losses

– Opportunity Losses


Stock too muchStock too much

Stock too fewStock too few


Random Variables





Combining Probabilities and Monetary Values

Extra Reading

Extra Reading


Possible Requests for Strawberries

Possible Stock Options

10 11 12 13

10 0 20 40 60

11 30 0 20 40

12 60 30 0 20

13 90 60 30 0

Obsolescence: Stock too muchObsolescence: Stock too much

Opportunity: Stock too fewOpportunity: Stock too few

$20 per case

$30 per case

30

30

30

60

90 60

20

20

20

40

40

60


Random Variables






Extra Reading

Extra Reading


Random Variables







10 1112

13

When n >=20 and p =< 0.05.

Equation: P.254


Random Variables





The Poisson ~Binomial Distribution

Extra Reading

Extra Reading

µ = n * pMean of a binomial distributionMean of a binomial distribution

σ = sqrt(npq)Standard deviation of a binomial distributionStandard deviation of a binomial distribution

Central Tendency and Dispersion of Binomial Distribution


Random Variables





Extra Reading

Extra Reading

Central Tendency and Dispersion of Binomial Distribution


Random Variables





20% defective packages10 packages

What are the mean and standard deviation?

Example:Example:

µ = n * p

=10*0.2 =2

σ = sqrt(npq)

=sqrt (10*0.2*0.8) =1.265

Extra Reading

Extra Reading

Lesson 1 Chapter 5 probability

Education

random variables

normal distribution

normal distribution

normal distribution

binomial distribution

poisson distribution

decision making

probability