chap 4-1 statistics for business and economics theory of probability

Chap 4-1Statistics for Business and Economics

Theory of Probability

Randomness

The term randomness suggests unpredictability A simple example of randomness is the tossing

of a coin. The outcome is uncertain; it can either be an observed head (H) or an observed tail (T). Because the outcome of the toss cannot be predicted for sure, we say it displays randomness.

Statistics for Business and Economics Chap 4-2

Uncertainty

At some time or another, everyone will experience uncertainty. For example, you are approaching the traffic signals and the light changes from green to amber. You have to decide whether you can make it trough the intersection or not. You may be uncertain as to what the correct decision should be.

Statistics for Business and EconomicsChap 4-3

Probability

The concept of probability is used to quantify this measure of doubt. If you believe that you have a 0.99 probability of getting across the intersection, you have made a clear statement about your doubt. The probability statement provides a great deal of information, much more than statements such as “Maybe I can make it across” or “I should make it across”



Important Definitions

Random Experiment – a process leading to an uncertain outcome

Basic Outcome – a possible outcome of a random experiment

Sample Space – the collection of all possible outcomes of a random experiment

Event – any subset of basic outcomes from the sample space



Intersection of Events – If A and B are two events in a sample space S, then the intersection, A ∩ B, is the set of all outcomes in S that belong to both A and B

(continued)

A BAB

S



A and B are Mutually Exclusive Events if they have no basic outcomes in common i.e., the set A ∩ B is empty

(continued)

A B

S



Union of Events – If A and B are two events in a sample space S, then the union, A U B, is the set of all outcomes in S that belong to either

A or B

(continued)

A B

The entire shaded area represents A U B

S



The Complement of an event A, , is the set of all basic outcomes in the sample space that do not belong to A.

(continued)

A

AS

A


Examples

Let the Sample Space be the collection of all possible outcomes of rolling one dice:

S = [1, 2, 3, 4, 5, 6]

Let A be the event “Number rolled is even”

Let B be the event “Number rolled is at least 4”

Then

A = [2, 4, 6] and B = [4, 5, 6]


(continued)

Examples

S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6]

5] 3, [1, A

6] [4, BA

6] 5, 4, [2, BA

S 6] 5, 4, 3, 2, [1, AA

Complements:

Intersections:

Unions:

[5] BA

3] 2, [1, B


Mutually exclusive: A and B are not mutually exclusive

The outcomes 4 and 6 are common to both

Collectively exhaustive: A and B are not collectively exhaustive

A U B does not contain 1 or 3

(continued)

Examples

S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6]


Probability

Probability – the chance that an uncertain event will occur

Mathematically, any function P(.) with

i. 0 ≤ P(A) ≤ 1

ii. P(ø)=0, P(S)=1

Certain

Impossible

.5

1

0

11iiA

iiAPP


Assessing Probability

There are three approaches to assessing the probability of an uncertain event:

1. classical probability

Assumes all outcomes in the sample space are equally likely to

occur

spacesampletheinoutcomesofnumbertotal

eventthesatisfythatoutcomesofnumber

N

NAeventofyprobabilit A


Counting the Possible Outcomes

Use the Combinations formula to determine the number of combinations of n things taken k at a time

where n! = n(n-1)(n-2)…(1) 0! = 1 by definition

k)!(nk!

n! Cn

k

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 4-16

Assessing Probability

Three approaches (continued)

2. relative frequency probability

the limit of the proportion of times that an event A occurs in a large number of trials, n

3. subjective probability

an individual opinion or belief about the probability of occurrence

populationtheineventsofnumbertotal

Aeventsatisfythatpopulationtheineventsofnumber

n

nAeventofyprobabilit A


Probability Distributions

Random Variable Represents a possible numerical value from

a random experiment

Random

Variables

Discrete Random Variable

ContinuousRandom Variable



Continuous Probability Distributions

Binomial

Hypergeometric

Poisson


Discrete Probability Distributions

Uniform

Normal

Exponential


Discrete Random Variables

Can only take on a countable number of values

Examples:

Roll a dice twiceLet X be the number of times 4 comes up (then X could be 0, 1, or 2 times)

Toss a coin 5 times. Let X be the number of heads

(then X could be 0, 1, 2, 3, 4, or 5)


Experiment: Toss 2 Coins. Let X = # heads.

T

T

Discrete Probability Distribution

4 possible outcomes

T

T

H

H

H H

Probability Distribution

0 1 2 x

x Value Probability

0 1/4 = .25

1 2/4 = .50

2 1/4 = .25

.50

.25

Pro

bab

ility

Show P(x) , i.e., P(X = x) , for all values of x:


The Normal Distribution

‘Bell Shaped’ Symmetrical Mean, Median and Mode

are Equal

Location is determined by the mean, μ

Spread is determined by the standard deviation, σ

The random variable has an infinite theoretical range: + to

Mean = Median = Mode

x

f(x)

μ

σ


The Normal Distribution

The normal distribution closely approximates the probability distributions of a wide range of random variables

Distributions of sample means approach a normal distribution given a “large” sample size

Computations of probabilities are direct and elegant

The normal probability distribution has led to good business decisions for a number of applications

(continued)


By varying the parameters μ and σ, we obtain different normal distributions

Many Normal Distributions


The Normal Distribution Shape

x

f(x)

μ

σ

Changing μ shifts the distribution left or right.

Changing σ increases or decreases the spread.

Given the mean μ and variance σ we define the normal distribution using the notation

)σN(μ~X 2,


The Normal Probability Density Function

The formula for the normal probability density function is

Where e = the mathematical constant approximated by 2.71828π = the mathematical constant approximated by 3.14159μ = the population meanσ = the population standard deviationx = any value of the continuous variable, < x <

22 /2σμ)(xe2π

1f(x)

Cumulative distribution function - CDF

Also called probability distribution function or just distribution function



Cumulative Probability Function

The cumulative probability function, denoted F(x0), shows the probability that X is less than or equal to x0

In other words,

)xP(X)F(x 00

0xx

0 P(x))F(x


Cumulative Normal Distribution

For a normal random variable X with mean μ and variance σ2 , i.e., X~N(μ, σ2), the cumulative distribution function is

)xP(X)F(x 00

x0 x0

)xP(X 0

f(x)

CDF vs. pdf

Cumulative distribution function is an integral of probability density function

Probability density function is a derivation of cumulative distribution function.

CDF is defined only for values of a random variable X that are less than a certain value (x). CDF represents the surface under the probability density function.



xbμa

The probability for a range of values is measured by the area under the curve

Finding Normal Probabilities

F(a)F(b)b)XP(a


xbμa

xbμa

xbμa

Finding Normal Probabilities (continued)

F(a)F(b)b)XP(a

a)P(XF(a)

b)P(XF(b)


Upper Tail Probabilities

Suppose X is normal with mean 8.0 and standard deviation 5.0.

Now Find P(X > 8.6)

X

8.6

8.0

P(X>b) 1 F(b)


The Standardized Normal

Any normal distribution (with any mean and variance combination) can be transformed into the standardized normal distribution (Z), with mean 0 and variance 1

Need to transform X units into Z units by subtracting the mean of X and dividing by its standard deviation

1)N(0~Z ,

σ

μXZ

Z

f(Z)

0

1


Example

If X is distributed normally with mean of 100 and standard deviation of 50, the Z value for X = 200 is

This says that X = 200 is two standard deviations (2 increments of 50 units) above the mean of 100.

2.050

100200

σ

μXZ


Comparing X and Z units

Z100

2.00200 X

Note that the distribution is the same, only the scale has changed. We can express the problem in original units (X) or in standardized units (Z)

(μ = 100, σ = 50)

(μ = 0, σ = 1)


Steps to find the X value for a known probability:1. Find the Z value for the known probability

2. Convert to X units using the formula:

Finding the X value for a Known Probability

ZσμX


Finding the X value for a known probability

Example: Suppose X is normal with mean 8.0 and

standard deviation 5.0. Now find the X value so that only 20% of all

values are below this X

X? 8.0

.2000

Z? 0

(continued)


Find the Z value for 20% in the Lower Tail

20% area in the lower tail is consistent with a Z value of -0.84

Standardized Normal Probability Table (Portion)

X? 8.0

.20

Z-0.84 0

1. Find the Z value for the known probability

z F(z)

.82 .7939

.83 .7967

.84 .7995

.85 .8023

.80


2. Convert to X units using the formula:

Finding the X value

80.3

0.5)84.0(0.8

ZσμX

So 20% of the values from a distribution with mean 8.0 and standard deviation 5.0 are less than 3.80


If the data distribution is bell-shaped, then the interval:

contains about 68% of the values in the population or the sample

The Empirical Rule

1σμ

μ

68%

1σμ


contains about 95% of the values in the population or the sample

contains about 99.7% of the values in the population or the sample

The Empirical Rule

2σμ

3σμ

3σμ

99.7%95%

2σμ

Next topic

Point and interval estimate

Thank you!

Have a nice day!

chap 4-1 statistics for business and economics theory of probability

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