1 Slide Slide Bases of the theory of probability and mathematical statistics.

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Bases of the theory of probability and mathematical statistics.

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An experiment is a situation involving chance or probability that leads to results called outcomes.In the problem above, the experiment is spinning the spinner.

An outcome is the result of a single trial of an experiment.The possible outcomes are landing on yellow, blue, green or red.

An event is one or more outcomes of an experiment.One event of this experiment is landing on blue.

Probability is the measure of how likely an event is.

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DefinitionsDefinitionsCertain

Impossible

.5

1

0

50/50

Probability is the numerical measure of the likelihood that the event will occur.

Value is between 0 and 1.

Sum of the probabilities of all eventsis 1.

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Experimental vs.Theoretical

Experimental probability:

P(event) = number of times event occurs

total number of trials

Theoretical probability:

P(E) = number of favorable outcomes total number of possible outcomes

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Identifying the Type of Probability

Trial Red Blue

1 1

2 1

3 1

4 1

5 1

6 1

Total 2 4

Exp. Prob. 1/3 2/3

You draw a marble out of the bag, record the color, and replace the marble. After 6 draws, you record 2 red marbles

P(red)= 2/6 = 1/3 Experimental(The result is found by

repeating an experiment.)

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The complement of A is everything in the sample space S that is NOT in A.

AS

•If the rectangular box is S, and the white circle is A, then everything in the box that’s outside the circle is Ac , which is the complement of A.

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TheoremPr (Ac) = 1 - Pr (A)

Example:

If A is the event that a randomly selected student is male, and the probability of A is 0.6, what is Ac and what is its probability?

Ac is the event that a randomly selected student is female, and its probability is 0.4.

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The union of A & B (denoted A U B) is everything in the sample space that is in either A or B or both.

AS

•The union of A & B is the whole white area.

B

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The intersection of A & B (denoted A∩B) is everything in the sample space that is in both A & B.

S

•The intersection of A & B is the pink overlapping area.

BA

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Example A family is planning to have 2 children. Suppose boys (B) & girls (G) are equally

likely.

What is the sample space S?

S = {BB, GG, BG, GB}


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Example continuedIf E is the event that both children are the same sex, what does E look like & what is its probability?

E = {BB, GG}

Since boys & girls are equally likely, each of the four outcomes in the sample space S = {BB, GG, BG, GB} is equally

likely & has a probability of 1/4.

So Pr(E) = 2/4 = 1/2 = 0.5


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Example cont’d: Recall that E = {BB, GG} & Pr(E)=0.5

What is the complement of E and what is its probability?

Ec = {BG, GB}

Pr (Ec) = 1- Pr(E) = 1 - 0.5 = 0.5


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Example continued

If F is the event that at least one of the children is a girl, what does F look like & what is its probability?

F = {BG, GB, GG}

Pr(F) = 3/4 = 0.75


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Recall: E = {BB, GG} & Pr(E)=0.5F = {BG, GB, GG} & Pr(F) = 0.75

What is E∩F? {GG}

What is its probability? 1/4 = 0.25


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Recall: E = {BB, GG} & Pr(E)=0.5F = {BG, GB, GG} & Pr(F) = 0.75What is the EUF?

{BB, GG, BG, GB} = SWhat is the probability of EUF?

1If you add the separate probabilities of E & F together, do you get Pr(EUF)? Let’s try it.Pr(E) + Pr(F) = 0.5 + 0.75 = 1.25 ≠ 1 = Pr (EUF)Why doesn’t it work?We counted GG (the intersection of E & F) twice.


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A formula for Pr(EUF)

Pr(EUF) = Pr(E) + Pr(F) - Pr(E∩F)

If E & F do not overlap, then the intersection is the empty set, & the probability of the intersection is zero.

When there is no overlap, Pr(EUF) = Pr(E) + Pr(F) .


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We can deduce an important result from the conditional law of probability:

( The probability of A does not depend on B. )

or P(A B) P(A) P(B)

If B has no effect on A, then, P(A B) = P(A) and we say the events are independent.

becomes P(A)

P(A B)P(B)

P(A|B)

So, P(A B)P(B)

Independent Independent EventsEvents


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Tests for independence

P(A B) P(A) P(B)

P(A B) P(A)

or

Independent Independent EventsEvents

P(B A) P(B)


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The Multiplication The Multiplication RuleRule

If events A and B are independent, then the probability of two events, A and B occurring in a sequence (or simultaneously) is:

This rule can extend to any number of independent events.

)()()()( BPAPBAPBandAP

Two events are independent if the occurrence of the first event does not affect the probability of the occurrence of the second event. More on this later


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Mutually Mutually ExclusiveExclusive

Two events A and B are mutually exclusive if and only if:

In a Venn diagram this means that event A is disjoint from event B.

A and B are M.E.

0)( BAP

A BA B

A and B are not M.E.


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The Addition The Addition RuleRule

The probability that at least one of the events A or B will occur, P(A or B), is given by:

If events A and B are mutually exclusive, then the addition rule is simplified to:

This simplified rule can be extended to any number of mutually exclusive events.

)()()()()( BAPBPAPBAPBorAP

)()()()( BPAPBAPBorAP


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Conditional Conditional ProbabilityProbability

Conditional probability is the probability of an event occurring, given that another event has already occurred. Conditional probability restricts the sample space.The conditional probability of event B occurring, given that event A has occurred, is denoted by P(B|A) and is read as “probability of B, given A.” We use conditional probability when two events occurring in sequence are not independent. In other words, the fact that the first event (event A) has occurred affects the probability that the second event (event B) will occur.


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Conditional Conditional ProbabilityProbability

Formula for Conditional Probability

Better off to use your brain and work out conditional probabilities from looking at the sample space, otherwise use the formula.

)(

)()|(

)(

)()|(

AP

ABPABPor

BP

BAPBAP


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Assigning ProbabilitiesAssigning Probabilities Two basic requirements for assigning probabilities

1. The probability assigned to each experimental outcome must be between 0 and 1, inclusively. If we let Ei denote the ith experimental outcome and P(Ei) its probability, then this requirement can be written as

0 P(Ei) 1 for all I

2. The sum of the probabilities for all the experimental outcomes must equal 1.0. For n experimental outcomes, this requirement can be written as

P(E1)+ P(E2)+… + P(En) =1


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Classical MethodClassical Method If an experiment has n possible outcomes, this method would assign a probability of 1/n to each outcome.

Experiment: Rolling a die

Sample Space: S = {1, 2, 3, 4, 5, 6}

Probabilities: Each sample point has a 1/6 chance of occurring

Example


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THEORETICAL PROBABILITY

I have a quarterMy quarter has a heads side and a tails side

Since my quarter has only 2 sides, there are only 2 possible outcomes when I flip it. It will either land on heads, or tails

HEADS

TAILS


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THEORETICAL PROBABILITY

When I flip my coin, the probability that my coin will land on heads is 1 in 2

What is the probability that my coin will land on tails??

HEADS

TAILS


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Theoretical Probability

Right!!! There is a 1 in 2 probability that my coin will land on tails!!!

HEADS

TAILS

A probability of 1 in 2 can be written in three ways:

•As a fraction: ½•As a decimal: .50

•As a percent: 50%


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I am going to take 1 marble from the bag. What is the probability that I will pick out

a red marble?


I have three marbles in a bag.

1 marble is red

1 marble is blue

1 marble is green


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Since there are three marbles and only one is red, I have a 1 in 3 chance of picking out a red marble.

I can write this in three ways:

As a fraction: 1/3 As a decimal: .33 As a percent: 33%


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Experimental Probability

Experimental probability is found by repeating an experiment and observing the outcomes.


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Remember the bag of marbles? The bag has only 1 red, 1 green,

and 1 blue marble in it. There are a total of 3 marbles in

the bag. Theoretical Probability says there

is a 1 in 3 chance of selecting a red, a green or a blue marble.


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Draw 1 marble from the bag.

Marble number red blue green

1 123456

It is a red marble.

Record the outcome on the tally sheet


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Put the red marble back in the bag and draw again.

This time your drew a green marble. Record this outcome on the tally sheet.


1 12 134


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Place the green marble back in the bag. Continue drawing marbles and recording

outcomes until you have drawn 6 times. (remember to place each marble back in the bag before drawing again.)


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After 6 draws your chart will look similar to this.

Look at the red column.

Of our 6 draws, we selected a red marble 2 times.


1 12 13 14 15 16 1

Total 2 1 3


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The experimental probability of drawing a red marble was 2 in 6.

This can be expressed as a fraction: 2/6 or 1/3 a decimal : .33 or a percentage: 33%


1 12 13 14 15 16 1

Total 2 1 3


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Notice the Experimental Probability of drawing a red, blue or green marble.


1 12 13 14 15 16 1

Total 2 1 3

Exp. Prob.

2/6 or 1/3 1/6

3/6 or 1/2


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Comparing Experimental and Theoretical Probability

Look at the chart at the right.

Is the experimental probability always the same as the theoretical probability?

red blue greenExp. Prob. 1/3 1/6 1/2Theo. Prob. 1/3 1/3 1/3


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In this experiment, the experimental and theoretical probabilities of selecting a red marble are equal.



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The experimental probability of selecting a blue marble is less than the theoretical probability.

The experimental probability of selecting a green marble is greater than the theoretical probability.



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Point and interval estimations of parameters of the normally up-diffused sign. Concept of statistical evaluation.


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What is statistics?

a branch of mathematics that provides techniques to analyze whether or not your data is significant (meaningful)

Statistical applications are based on probability statements

Nothing is “proved” with statistics Statistics are reported Statistics report the probability that similar

results would occur if you repeated the experiment


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Statistics deals with numbers

Need to know nature of numbers collectedContinuous variables: type of numbers associated

with measuring or weighing; any value in a continuous interval of measurement. Examples:

Weight of students, height of plants, time to flowering

Discrete variables: type of numbers that are counted or categorical Examples:

Numbers of boys, girls, insects, plants


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Standard Deviation and Variance Standard deviation and variance are the

most common measures of total risk

They measure the dispersion of a set of observations around the mean observation


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Standard Deviation and Variance (cont’d) General equation for variance:

If all outcomes are equally likely:

2

2

1

Variance prob( )n

i ii

x x x

2

2

1

1 n

ii

x xn


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Standard Deviation and Variance (cont’d) Equation for standard deviation:

2

2

1

Standard deviation prob( )n

i ii

x x x


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1.The Normal distribution – parameters and (or 2)

Comment: If = 0 and = 1 the distribution is called the standard normal distribution

0

0.005

0.01

0.015

0.02

0.025

0.03

0 20 40 60 80 100 120

Normal distribution with = 50 and =15

Normal distribution with = 70 and =20


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( ) e ,2

x

f x x

The probability density of the normal distribution

If a random variable, X, has a normal distribution with mean and variance 2 then we will write:

2~ ,X N


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The Chi-square (2) distribution with d.f.

21

2 2

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2

0

0 0

xx e xf x

x

2

2 2

22

10

2

0 0

x

x e x

x

The Chi-square distribution


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0

0.1

0.2

0 4 8 12 16

Graph: The 2 distribution

( = 4)

( = 5)

( = 6)


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1. If z has a Standard Normal distribution then z2

has a 2 distribution with 1 degree of freedom.

Basic Properties of the Chi-Square distribution

2. If z1, z2,…, z are independent random variables each having Standard Normal distribution then

has a 2 distribution with degrees of freedom.

2 2 21 2 ...U z z z

3. Let X and Y be independent random variables having a 2 distribution with 1 and 2 degrees of freedom respectively then X + Y has a 2

distribution with degrees of freedom 1 + 2.


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continued

4. Let x1, x2,…, xn, be independent random variables having a 2 distribution with 1 , 2 ,…, n degrees of freedom respectively then x1+ x2 +…+ xn has a 2 distribution with degrees of freedom 1 +…+ n.

5. Suppose X and Y are independent random variables with X and X + Y having a 2 distribution with 1

and ( > 1 ) degrees of freedom respectively then Y has a 2 distribution with degrees of freedom - 1.


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The non-central Chi-squared distribution

If z1, z2,…, z are independent random variables each having a Normal distribution with mean i and variance 2 = 1, then

has a non-central 2 distribution with degrees of freedom and non-centrality parameter

2 2 21 2 ...U z z z

1

221

ii


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Mean and Variance of non-central 2 distribution

If U has a non-central 2 distribution with degrees of freedom and non-centrality parameter

1

221

ii

Then

1

22i

iUE 42 UVar

If U has a central 2 distribution with degrees of freedom and is zero, thus

UE 2UVar


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Estimation of Population Parameters

Statistical inference refers to making inferences about a population parameter through the use of sample information

The sample statistics summarize sample information and can be used to make inferences about the population parameters

Two approaches to estimate population parameters Point estimation: Obtain a value estimate for the population parameter Interval estimation: Construct an interval within which the population

parameter will lie with a certain probability


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Point Estimation

In attempting to obtain point estimates of population parameters, the following questions arise What is a point estimate of the population mean? How good of an estimate do we obtain through the methodology that we

follow?

Example: What is a point estimate of the average yield on ten-year Treasury bonds?

To answer this question, we use a formula that takes sample information and produces a number


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Point Estimation

A formula that uses sample information to produce an estimate of a population parameter is called an estimator

A specific value of an estimator obtained from information of a specific sample is called an estimate

Example: We said that the sample mean is a good estimate of the population mean The sample mean is an estimator A particular value of the sample mean is an estimate


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Interval Estimation

In the probabilistic interpretation, we say that

A 95% confidence interval for a population parameter means that, in repeated sampling, 95% of such confidence intervals will include the population parameter

In the practical interpretation, we say that

We are 95% confident that the 95% confidence interval will include the population parameter


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Constructing Confidence Intervals

Confidence intervals have similar structures

Point Estimate Reliability Factor Standard Error

Reliability factor is a number based on the assumed distribution of the point estimate and the level of confidence

Standard error of the sample statistic providing the point estimate


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Confidence Interval for Mean of a Normal Distribution with Known Variance

If is the sample mean, then we are interested in the confidence interval, such that the following probability is .9

X

nX

nXP

nX

nP

n

XP

ZP

645.1645.1

645.1645.1

645.1/

645.1

645.1645.19.


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Following the above expression for the structure of a confidence interval, we rewrite the confidence interval as follows

Note that from the standard normal density

nX

645.1

05.065.1)65.1(

95.065.165.1

Z

Z

FZP

FZP


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Given this result and that the level of confidence for this interval (1-) is .90, we conclude that

The area under the standard normal to the left of –1.65 is 0.05 The area under the standard normal to the right of 1.65 is 0.05

Thus, the two reliability factors represent the cutoffs -z/2 and z/2 for the standard normal


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In general, a 100(1-)% confidence interval for the population mean when we draw samples from a normal distribution with known variance 2 is given by

where z/2 is the number for which

nzX

2/

22/

zZP


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Note: We typically use the following reliability factors when constructing confidence intervals based on the standard normal distribution

90% interval: z0.05 = 1.65

95% interval: z0.025 = 1.96

99% interval: z0.005 = 2.58

Implication: As the degree of confidence increases the interval becomes wider


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Example: Suppose we draw a sample of 100 observations of returns on the Nikkei index, assumed to be normally distributed, with sample mean 4% and standard deviation 6%

What is the 95% confidence interval for the population mean?

The standard error is .06/ = .006

The confidence interval is .04 1.96(.006)

100


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Confidence Interval for Mean of a Normal Distribution with Unknown Variance

In a more typical scenario, the population variance is unknown

Note that, if the sample size is large, the previous results can be modified as follows The population distribution need not be normal The population variance need not be known The sample standard deviation will be a sufficiently good estimator of

the population standard deviation

Thus, the confidence interval for the population mean derived above can be used by substituting s for


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However, if the sample size is small and the population variance is unknown, we cannot use the standard normal distribution

If we replace the unknown with the sample st. deviation s the following quantity

follows Student’s t distribution with (n – 1) degrees of freedom

ns

Xt

/


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The t-distribution has mean 0 and (n – 1) degrees of freedom

As degrees of freedom increase, the t-distribution approaches the standard normal distribution

Also, t-distributions have fatter tails, but as degrees of freedom increase (df = 8 or more) the tails become less fat and resemble that of a normal distribution


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In general, a 100(1-)% confidence interval for the population mean when we draw small samples from a normal distribution with an unknown variance 2 is given by

where tn-1,/2 is the number for which

n

stX n 2/,1

22/,11

nn ttP


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Confidence Interval for the Population Variance of a Normal Population

Suppose we have obtained a random sample of n observations from a normal population with variance 2 and that the sample variance is s2. A 100(1 - )% confidence interval for the population variance is

2

2/1,1

22

22/,1

2 11

nn

snsn


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1 Slide Slide Bases of the theory of probability and mathematical statistics.

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